{"id":15554,"date":"2019-09-05T12:07:57","date_gmt":"2019-09-05T16:07:57","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/hyperbolas-2\/"},"modified":"2019-09-05T12:07:57","modified_gmt":"2019-09-05T16:07:57","slug":"hyperbolas-2","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/hyperbolas-2\/","title":{"raw":"Hyperbolas","rendered":"Hyperbolas"},"content":{"raw":"[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Graph a hyperbola with center at \\(\\left(0,0\\right)\\)<\/li><li>Graph a hyperbola with center at \\(\\left(h,k\\right)\\)<\/li><li>Identify conic sections by their equations<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1163873928434\" class=\"be-prepared\"><p id=\"fs-id1163873605884\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1163873624750\" type=\"1\"><li>Solve: \\({x}^{2}=12.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/b9659e42-3afa-4449-81d9-a017c35de140#fs-id1167834228017\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Expand: \\({\\left(x-4\\right)}^{2}.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Graph \\(y=-\\frac{2}{3}x.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/26e8f94c-1f76-46ec-8e6c-344f06971cf5#fs-id1167834408393\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163870348934\"><h3 data-type=\"title\">Graph a Hyperbola with Center at <strong data-effect=\"bold\">(0, 0)<\/strong><\/h3><p id=\"fs-id1163873616698\">The last conic section we will look at is called a <span data-type=\"term\">hyperbola<\/span>. We will see that the equation of a hyperbola looks the same as the equation of an ellipse, except it is a difference rather than a sum. While the equations of an ellipse and a hyperbola are very similar, their graphs are very different.<\/p><p id=\"fs-id1163870487157\">We define a <strong data-effect=\"bold\">hyperbola<\/strong> as all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the hyperbola.<\/p><div data-type=\"note\" id=\"fs-id1163873631003\"><div data-type=\"title\">Hyperbola<\/div><p id=\"fs-id1163873641928\">A <strong data-effect=\"bold\">hyperbola<\/strong> is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the hyperbola.<\/p><span data-type=\"media\" id=\"fs-id1163873648722\" data-alt=\"The figure shows a double napped right circular cone sliced by a plane that is parallel to the vertical axis of the cone forming a hyperbola. The figure is labeled &#x2018;hyperbola&#x2019;.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a double napped right circular cone sliced by a plane that is parallel to the vertical axis of the cone forming a hyperbola. The figure is labeled &#x2018;hyperbola&#x2019;.\" \/><\/span><\/div><p id=\"fs-id1163873644949\">The line through the foci, is called the <strong data-effect=\"bold\">transverse axis<\/strong>. The two points where the transverse axis intersects the hyperbola are each a <strong data-effect=\"bold\">vertex<\/strong> of the hyperbola. The midpoint of the segment joining the foci is called the <strong data-effect=\"bold\">center<\/strong> of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the <strong data-effect=\"bold\">conjugate axis<\/strong>. Each piece of the graph is called a <strong data-effect=\"bold\">branch<\/strong> of the hyperbola.<\/p><span data-type=\"media\" id=\"fs-id1163870489434\" data-alt=\"The figure shows two graphs of a hyperbola. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci are shown with points that lie on the transverse axis, which is the x-axis. The branches pass through the vertices and open left and right. The y-axis is the conjugate axis. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci lie are shown with points that lie on the transverse axis, which is the y-axis. The branches pass through the vertices and open up and down. The x-axis is the conjugate axis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows two graphs of a hyperbola. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci are shown with points that lie on the transverse axis, which is the x-axis. The branches pass through the vertices and open left and right. The y-axis is the conjugate axis. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci lie are shown with points that lie on the transverse axis, which is the y-axis. The branches pass through the vertices and open up and down. The x-axis is the conjugate axis.\" \/><\/span><p id=\"fs-id1163873631136\">Again our goal is to connect the geometry of a conic with algebra. Placing the hyperbola on a rectangular coordinate system gives us that opportunity. In the figure, we placed the hyperbola so the foci \\(\\left(\\left(\\text{\u2212}c,0\\right),\\left(c,0\\right)\\right)\\) are on the <em data-effect=\"italics\">x<\/em>-axis and the center is the origin.<\/p><span data-type=\"media\" id=\"fs-id1163873622834\" data-alt=\"The figure shows the graph of a hyperbola. The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The foci (negative c, 0) and (c, 0) are marked with a point and lie on the x-axis. The vertices are marked with a point and lie on the x-axis. The branches pass through the vertices and open left and right. The distance from (negative c, 0) to a point on the branch (x, y) is marked d sub 1. The distance from (x, y) on the branch to (c, 0) is marked d sub 2.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a hyperbola. The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The foci (negative c, 0) and (c, 0) are marked with a point and lie on the x-axis. The vertices are marked with a point and lie on the x-axis. The branches pass through the vertices and open left and right. The distance from (negative c, 0) to a point on the branch (x, y) is marked d sub 1. The distance from (x, y) on the branch to (c, 0) is marked d sub 2.\" \/><\/span><p id=\"fs-id1163873764219\">The definition states the difference of the distance from the foci to a point \\(\\left(x,y\\right)\\) is constant. So \\(|{d}_{1}-{d}_{2}|\\) is a constant that we will call \\(2a\\) so \\(|{d}_{1}-{d}_{2}|=2a.\\) We will use the distance formula to lead us to an algebraic formula for an ellipse.<\/p><p id=\"fs-id1163873857481\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}|{d}_{1}\\phantom{\\rule{3.5em}{0ex}}-\\phantom{\\rule{3.5em}{0ex}}{d}_{2}|\\phantom{\\rule{3.2em}{0ex}}=2a\\\\ \\text{Use the distance formula to find}\\phantom{\\rule{0.2em}{0ex}}{d}_{1},{d}_{2}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}|\\sqrt{{\\left(x-\\left(-c\\right)\\right)}^{2}+{\\left(y-0\\right)}^{2}}-\\sqrt{{\\left(x-c\\right)}^{2}+{\\left(y-0\\right)}^{2}}\\phantom{\\rule{0.2em}{0ex}}|=2a\\\\ \\text{Eliminate the radicals.}\\hfill &amp; &amp; &amp; \\\\ \\begin{array}{c}\\text{To simplify the equation of the ellipse, we}\\hfill \\\\ \\text{let}\\phantom{\\rule{0.2em}{0ex}}{c}^{2}-{a}^{2}={b}^{2}.\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{c}^{2}-{a}^{2}}=1\\phantom{\\rule{0.53em}{0ex}}\\\\ \\begin{array}{c}\\text{So, the equation of a hyperbola centered at}\\hfill \\\\ \\text{the origin in standard form is:}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1\\phantom{\\rule{0.53em}{0ex}}\\end{array}\\)<\/p><p id=\"fs-id1163873672757\">To graph the hyperbola, it will be helpful to know about the intercepts. We will find the <em data-effect=\"italics\">x<\/em>-intercepts and <em data-effect=\"italics\">y<\/em>-intercepts using the formula.<\/p><p id=\"fs-id1163873799747\">\\(\\begin{array}{c}\\begin{array}{cccccccccccccccc}&amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}{\\text{x}}\\mathbf{\\text{-intercepts}}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{9em}{0ex}}{\\text{y}}\\mathbf{\\text{-intercepts}}\\hfill \\end{array}\\hfill \\\\ \\begin{array}{cccccccccccccccccccc}&amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ \\text{Let}\\phantom{\\rule{0.2em}{0ex}}y=0.\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{a}^{2}}-\\frac{{0}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}\\text{Let}\\phantom{\\rule{0.2em}{0ex}}x=0.\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{0}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill \\frac{{x}^{2}}{{a}^{2}}&amp; =\\hfill &amp; 1\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill -\\frac{{y}^{2}}{{b}^{2}}&amp; =\\hfill &amp; 1\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill {x}^{2}&amp; =\\hfill &amp; {a}^{2}\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill {y}^{2}&amp; =\\hfill &amp; \\text{\u2212}{b}^{2}\\hfill \\\\ &amp; &amp; &amp; &amp; &amp; \\hfill x&amp; =\\hfill &amp; \\text{\u00b1}a\\hfill &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; &amp; \\hfill y&amp; =\\hfill &amp; \\text{\u00b1}\\sqrt{\\text{\u2212}{b}^{2}}\\hfill \\end{array}\\hfill \\\\ \\begin{array}{cccccc}\\phantom{\\rule{2em}{0ex}}\\text{The}\\phantom{\\rule{0.2em}{0ex}}x\\text{-intercepts are}\\phantom{\\rule{0.2em}{0ex}}\\left(a,0\\right)\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}\\left(\\text{\u2212}a,0\\right).\\hfill &amp; &amp; &amp; &amp; &amp; \\phantom{\\rule{5em}{0ex}}\\text{There are no}\\phantom{\\rule{0.2em}{0ex}}y\\text{-intercepts.}\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1163873800186\">The <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em> values in the equation also help us find the asymptotes of the hyperbola. The asymptotes are intersecting straight lines that the branches of the graph approach but never intersect as the <em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em> values get larger and larger.<\/p><p id=\"fs-id1163873646422\">To find the asymptotes, we sketch a rectangle whose sides intersect the <em data-effect=\"italics\">x<\/em>-axis at the vertices \\(\\left(\\text{\u2212}a,0\\right),\\) \\(\\left(a,0\\right)\\) and intersect the <em data-effect=\"italics\">y<\/em>-axis at \\(\\left(0,\\text{\u2212}b\\right),\\) \\(\\left(0,b\\right).\\) The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola. The rectangle and asymptotes are not part of the hyperbola, but they help us graph the hyperbola.<\/p><span data-type=\"media\" id=\"fs-id1163873809752\" data-alt=\"The figure shows the graph of a hyperbola. The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (negative a, 0) and (a, 0) and are marked with a point and lie on the x-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the x-axis at the vertices (negative a, 0) and (a, 0) and intersect the y-axis at (0, b) and (0, negative b). The asymptotes are given by y is equal to b divided by a times x and y is equal to negative b divided by a times x and are drawn as the diagonals of the central rectangle. The branches of the hyperbola pass through the vertices, open left and right, and approach the asymptotes.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a hyperbola. The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (negative a, 0) and (a, 0) and are marked with a point and lie on the x-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the x-axis at the vertices (negative a, 0) and (a, 0) and intersect the y-axis at (0, b) and (0, negative b). The asymptotes are given by y is equal to b divided by a times x and y is equal to negative b divided by a times x and are drawn as the diagonals of the central rectangle. The branches of the hyperbola pass through the vertices, open left and right, and approach the asymptotes.\" \/><\/span><p id=\"fs-id1163869199949\">The asymptotes pass through the origin and we can evaluate their slope using the rectangle we sketched. They have equations \\(y=\\frac{b}{a}x\\) and \\(y=-\\frac{b}{a}x.\\)<\/p><p id=\"fs-id1163873510103\">There are two equations for hyperbolas, depending whether the transverse axis is vertical or horizontal. We can tell whether the transverse axis is horizontal by looking at the equation. When the equation is in standard form, if the <em data-effect=\"italics\">x<\/em><sup>2<\/sup>-term is positive, the transverse axis is horizontal. When the equation is in standard form, if the <em data-effect=\"italics\">y<\/em><sup>2<\/sup>-term is positive, the transverse axis is vertical.<\/p><p id=\"fs-id1163873744316\">The second equations could be derived similarly to what we have done. We will summarize the results here.<\/p><div data-type=\"note\" id=\"fs-id1163873858397\"><div data-type=\"title\">Standard Form of the Equation a Hyperbola with Center \\(\\left(0,0\\right)\\)<\/div><p id=\"fs-id1163873645567\">The standard form of the equation of a hyperbola with center \\(\\left(0,0\\right),\\) is<\/p><div data-type=\"equation\" id=\"fs-id1163873668215\" class=\"unnumbered\" data-label=\"\">\\(\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1\\phantom{\\rule{2em}{0ex}}\\text{or}\\phantom{\\rule{2em}{0ex}}\\frac{{y}^{2}}{{a}^{2}}-\\frac{{x}^{2}}{{b}^{2}}=1\\)<\/div><span data-type=\"media\" id=\"fs-id1163870621360\" data-alt=\"The figure shows the graph of two hyperbolas. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (negative a, 0) and (a, 0) and are marked with a point and lie on the x-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the x-axis at the vertices (negative a, 0) and (a, 0) and intersect the y-axis at (0, b) and (0, negative b). The asymptotes are given by y is equal to b divided by a times x and y is equal to negative b divided by a times x and are drawn as the diagonals of the central rectangle. The branches of the hyperbola pass through the vertices, open left and right, and approach the asymptotes. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (0, a) and (0, negative a) and are marked with a point and lie on the y-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the y-axis at the vertices (0, a) and (0, negative a) and intersect the y-axis at (negative b, 0) and (b, 0). The branches of the hyperbola pass through the vertices, open up and down, and approach the asymptotes.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_005_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of two hyperbolas. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (negative a, 0) and (a, 0) and are marked with a point and lie on the x-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the x-axis at the vertices (negative a, 0) and (a, 0) and intersect the y-axis at (0, b) and (0, negative b). The asymptotes are given by y is equal to b divided by a times x and y is equal to negative b divided by a times x and are drawn as the diagonals of the central rectangle. The branches of the hyperbola pass through the vertices, open left and right, and approach the asymptotes. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (0, a) and (0, negative a) and are marked with a point and lie on the y-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the y-axis at the vertices (0, a) and (0, negative a) and intersect the y-axis at (negative b, 0) and (b, 0). The branches of the hyperbola pass through the vertices, open up and down, and approach the asymptotes.\" \/><\/span><p id=\"fs-id1163873663161\">Notice that, unlike the equation of an ellipse, the denominator of \\({x}^{2}\\) is not always \\({a}^{2}\\) and the denominator of \\({y}^{2}\\) is not always \\({b}^{2}.\\)<\/p><p id=\"fs-id1163873724345\">Notice that when the \\({x}^{2}\\)-term is positive, the transverse axis is on the <em data-effect=\"italics\">x<\/em>-axis. When the \\({y}^{2}\\)-term is positive, the transverse axis is on the <em data-effect=\"italics\">y<\/em>-axis.<\/p><\/div><table id=\"fs-id1163873605544\" summary=\"The table has three columns and eight rows. The first row is a title row and is labeled &#x2018;Standard Forms of the Equation a Hyperbola with Center (0, 0). The second row is a header row with the first column labeled the quantity x squared divided by a squared end quantity minus the quantity y squared divided by b squared is equal to 1 and the second column labeled the quantity y squared divided by a squared end quantity minus the quantity x squared divided by b squared is equal to 1. The rows are labeled &#x2018;Orientation&#x2019;, &#x2018;Vertices&#x2019;, &#x2018;x-intercepts&#x2019;, &#x2018;y-intercepts&#x2019;, &#x2018;Rectangle&#x2019;, and &#x2018;Asymptotes. In row three, the &#x2018;Orientations&#x2019; are &#x2018;transverse axis on the x-axis; opens left and right&#x2019; and &#x2018;transverse axis on the y-axis; opens up and down&#x2019;. In row four, the &#x2018;Vertices&#x2019; are (negative a, 0) and (a, 0) and (0, negative a) and (0, a). In row five, the x-intercepts are (negative a, 0) and (a, 0) and &#x2018;none). In row six, the y-intercepts are &#x2018;none&#x2019; and (0, negative a) and (0, a). In row seven, the rectangle uses (plus or minus a, 0) and (0, plus or minus) and uses (0, plus or minus a) and (plus or minus b, 0). In row eight, the asymptotes are y is equal to b divided by a times x, y is equal to negative b divided by a times x and y is equal to a divided by b times x and y is equal to negative a divided by b times x.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Standard Forms of the Equation a Hyperbola with Center \\(\\left(0,0\\right)\\)<\/th><\/tr><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"center\"><\/th><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1\\)<\/th><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{y}^{2}}{{a}^{2}}-\\frac{{x}^{2}}{{b}^{2}}=1\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Orientation<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Transverse axis on the <em data-effect=\"italics\">x<\/em>-axis.<span data-type=\"newline\"><br \/><\/span>Opens left and right<\/td><td data-valign=\"middle\" data-align=\"center\">Transverse axis on the <em data-effect=\"italics\">y<\/em>-axis.<span data-type=\"newline\"><br \/><\/span>Opens up and down<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Vertices<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(\\text{\u2212}a,0\\right),\\)\\(\\left(a,0\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,\\text{\u2212}a\\right),\\)\\(\\left(0,a\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(\\text{\u2212}a,0\\right),\\)\\(\\left(a,0\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"center\">none<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">none<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,\\text{\u2212}a\\right),\\)\\(\\left(0,a\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Rectangle<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Use \\(\\left(\\text{\u00b1}a,0\\right)\\) \\(\\left(0,\\text{\u00b1}b\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Use \\(\\left(0,\\text{\u00b1}a\\right)\\) \\(\\left(\\text{\u00b1}b,0\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">asymptotes<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(y=\\frac{b}{a}x,\\)\\(y=-\\frac{b}{a}x\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(y=\\frac{a}{b}x,\\)\\(y=-\\frac{a}{b}x\\)<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1163873552620\">We will use these properties to graph hyperbolas.<\/p><div data-type=\"example\" id=\"fs-id1163873626915\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Graph a Hyperbola with Center \\(\\left(0,0\\right)\\)<\/div><div data-type=\"exercise\" id=\"fs-id1163873626917\"><div data-type=\"problem\" id=\"fs-id1163873626919\"><p id=\"fs-id1163873800346\">Graph \\(\\frac{{x}^{2}}{25}-\\frac{{y}^{2}}{4}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873603471\"><span data-type=\"media\" id=\"fs-id1163873603473\" data-alt=\"Step 1 is to write the equation in standard form. The the quantity x squared divided by 25 end quantity minus the quantity y squared divided by 4 end quantity is equal to 1 is already in standard form.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the equation in standard form. The the quantity x squared divided by 25 end quantity minus the quantity y squared divided by 4 end quantity is equal to 1 is already in standard form.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873814884\" data-alt=\"Step 2 is to determine whether the transverse axis is horizontal or vertical. Since the x squared term is positive, the transverse axis is horizontal.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to determine whether the transverse axis is horizontal or vertical. Since the x squared term is positive, the transverse axis is horizontal.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873606120\" data-alt=\"Step 3 is to find the vertices. Since a squared is equal to 25, then a is equal to plus or minus 5. The vertices lie on the x-axis and are (negative 5, 0) and (5, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to find the vertices. Since a squared is equal to 25, then a is equal to plus or minus 5. The vertices lie on the x-axis and are (negative 5, 0) and (5, 0).\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873899041\" data-alt=\"Step 4 is to sketch the rectangle centered at the origin, intersecting one axis at plus or minus a and the other at plus or minus b. Since a is equal to plus or minus 5, the rectangle will intersect the x-axis at the vertices. Since b is equal to plus or minus 2, the rectangle will intersect the y-axis at (0, negative 2) and (0, 2). The rectangle is shown on a coordinate plane with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to sketch the rectangle centered at the origin, intersecting one axis at plus or minus a and the other at plus or minus b. Since a is equal to plus or minus 5, the rectangle will intersect the x-axis at the vertices. Since b is equal to plus or minus 2, the rectangle will intersect the y-axis at (0, negative 2) and (0, 2). The rectangle is shown on a coordinate plane with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873508574\" data-alt=\"Step 5 is to sketch the asymptotes, the lines through the diagonals of the rectangle. The asymptotes have the equations y is equal to five-halves times x and y is equal to negative five-halves x. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled and the lines that represent the asymptotes.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to sketch the asymptotes, the lines through the diagonals of the rectangle. The asymptotes have the equations y is equal to five-halves times x and y is equal to negative five-halves x. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled and the lines that represent the asymptotes.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873676691\" data-alt=\"Step 6 is to draw the two branches of the hyperbola. Start at each vertex and use the asymptotes as a guide. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled, the lines that represent the asymptotes, y is equal to plus or minus five-halves times x, and the branches that pass through (plus or minus 5, 0) and open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to draw the two branches of the hyperbola. Start at each vertex and use the asymptotes as a guide. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled, the lines that represent the asymptotes, y is equal to plus or minus five-halves times x, and the branches that pass through (plus or minus 5, 0) and open left and right.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873758424\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873758427\"><div data-type=\"problem\" id=\"fs-id1163873770598\"><p id=\"fs-id1163873770600\">Graph \\(\\frac{{x}^{2}}{16}-\\frac{{y}^{2}}{4}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873632906\"><span data-type=\"media\" id=\"fs-id1163873632910\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873629403\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873629407\"><div data-type=\"problem\" id=\"fs-id1163870357696\"><p id=\"fs-id1163870357699\">Graph \\(\\frac{{x}^{2}}{9}-\\frac{{y}^{2}}{16}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873644840\"><span data-type=\"media\" id=\"fs-id1163873644843\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus four-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus four-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163873893175\">We summarize the steps for reference.<\/p><div data-type=\"note\" id=\"fs-id1163873893179\" class=\"howto\"><div data-type=\"title\">Graph a hyperbola centered at \\(\\left(0,0\\right).\\)<\/div><ol id=\"fs-id1163873853922\" type=\"1\" class=\"stepwise\"><li>Write the equation in standard form.<\/li><li>Determine whether the transverse axis is horizontal or vertical.<\/li><li>Find the vertices.<\/li><li>Sketch the rectangle centered at the origin intersecting one axis at \\(\\text{\u00b1}a\\) and the other at \\(\\text{\u00b1}b.\\)<\/li><li>Sketch the asymptotes\u2014the lines through the diagonals of the rectangle.<\/li><li>Draw the two branches of the hyperbola.<\/li><\/ol><\/div><p id=\"fs-id1163873666021\">Sometimes the equation for a hyperbola needs to be first placed in standard form before we graph it.<\/p><div data-type=\"example\" id=\"fs-id1163873666024\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873557621\"><div data-type=\"problem\" id=\"fs-id1163873557623\"><p id=\"fs-id1163873557625\">Graph \\(4{y}^{2}-16{x}^{2}=64.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870557545\"><table id=\"fs-id1163873862986\" class=\"unnumbered unstyled\" summary=\"4 y squared minus 16 x squared is equal to 64. To write the equation in standard form, divide each term by 64 to make the equation equal to 1. The quantity 4 y divided by 64 end quantity minus 16 x divided by 64 end quantity is equal to the quantity 64 divided by 64. Simplify. The result is the quantity y squared divided by 16 end quantity minus the quantity x squared divided by 4 end quantity is equal to 1. Since the y-squared term is positive, the transverse axis is vertical. Since a squared is equal to 16, then a is equal to plus or minus 4. The vertices lie on the y-axis and are (0, negative a) and (0, a). The vertices are (0, negative 4) and (0, 4). Since b squared is equal to 4, then b is equal to plus or minus 2. Sketch the rectangle intersecting the x-axis at (negative 2, 0) and (2, 0) and the y-axis at the vertices. Sketch the asymptotes through the diagonals of the rectangle. Draw the two branches of the hyperbola. The graph that results is a rectangle that intersects the x-axis at (plus or minus 2, 0) and the y-axis at (0, plus or minus 4), asymptotes that are the diagonals are the rectangle, and branches that pass through the vertices (0, plus or minus 4), and that open up and down.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"center\">\\(4{y}^{2}-16{x}^{2}=64\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To write the equation in standard form, divide<span data-type=\"newline\"><br \/><\/span>each term by 64 to make the equation equal to 1.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\frac{4{y}^{2}}{64}-\\frac{16{x}^{2}}{64}=\\frac{64}{64}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\phantom{\\rule{1em}{0ex}}\\frac{{y}^{2}}{16}-\\frac{{x}^{2}}{4}=1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since the <em data-effect=\"italics\">y<\/em><sup>2<\/sup>-term is positive, the transverse axis is vertical.<span data-type=\"newline\"><br \/><\/span>Since \\({a}^{2}=16\\) then \\(a=\\text{\u00b1}4.\\)<\/td><td data-valign=\"top\" data-align=\"left\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">The vertices are on the <em data-effect=\"italics\">y<\/em>-axis, \\(\\left(0,\\text{\u2212}a\\right),\\) \\(\\left(0,a\\right).\\)<span data-type=\"newline\"><br \/><\/span>Since \\({b}^{2}=4\\) then \\(b=\\text{\u00b1}2.\\)<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\left(0,-4\\right),\\)\\(\\left(0,4\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the rectangle intersecting the <em data-effect=\"italics\">x<\/em>-axis at \\(\\left(-2,0\\right),\\) \\(\\left(2,0\\right)\\) and the <em data-effect=\"italics\">y<\/em>-axis at the vertices.<span data-type=\"newline\"><br \/><\/span>Sketch the asymptotes through the diagonals of the rectangle.<span data-type=\"newline\"><br \/><\/span>Draw the two branches of the hyperbola.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869153025\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163870619489\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163870619492\"><div data-type=\"problem\" id=\"fs-id1163873819474\"><p id=\"fs-id1163873819476\">Graph \\(4{y}^{2}-25{x}^{2}=100.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873604880\"><span data-type=\"media\" id=\"fs-id1163873604883\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163870504871\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163870504874\"><div data-type=\"problem\" id=\"fs-id1163870504876\"><p id=\"fs-id1163870504878\">Graph \\(25{y}^{2}-9{x}^{2}=225.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873800252\"><span data-type=\"media\" id=\"fs-id1163873800255\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus three-fifths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus three-fifths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\" \/><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873912654\"><h3 data-type=\"title\">Graph a Hyperbola with Center at \\(\\left(h,k\\right)\\)<\/h3><p id=\"fs-id1163874047291\">Hyperbolas are not always centered at the origin. When a hyperbola is centered at \\(\\left(h,k\\right)\\) the equations changes a bit as reflected in the table.<\/p><table id=\"fs-id1163873998386\" summary=\"The table has three columns and six rows. The first row is a title row and is labeled &#x2018;Standard Forms of the Equation a Hyperbola with Center (h, k). The second row is a header row with the first column labeled the quantity x minus h squared all divided by a squared end quantity minus the quantity y minus k squared all divided by b squared end quantity is equal to 1 and the second column labeled the quantity y minus k squared all divided by a squared end quantity minus the quantity x minus h squared all divided by b squared end quantity is equal to 1. The rows are labeled &#x2018;Orientation&#x2019;, &#x2018;Center&#x2019;, &#x2018;Vertices&#x2019;, and &#x2018;Rectangle&#x2019;. In row three, the &#x2018;Orientations&#x2019; are &#x2018;transverse axis on the x-axis; opens left and right&#x2019; and &#x2018;transverse axis on the y-axis; opens up and down&#x2019;. In row four, the &#x2018;Centers&#x2019; are both (h, k). In row five, the &#x2018;Vertices&#x2019; are a units to the left and right of the center and a units above and below the center. In row six, the &#x2018;Rectangles&#x2019; are formed by moving a units left or right of the center and b units above or below the center, and by using a units above or below the center and b units left or right of the center.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Standard Forms of the Equation a Hyperbola with Center \\(\\left(h,k\\right)\\)<\/th><\/tr><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"center\"><\/th><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1\\)<\/th><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}=1\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Orientation<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Transverse axis is horizontal.<span data-type=\"newline\"><br \/><\/span>Opens left and right<\/td><td data-valign=\"middle\" data-align=\"center\">Transverse axis is vertical.<span data-type=\"newline\"><br \/><\/span>Opens up and down<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Center<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(h,k\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(h,k\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Vertices<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><em data-effect=\"italics\">a<\/em> units to the left and right of the center<\/td><td data-valign=\"middle\" data-align=\"center\"><em data-effect=\"italics\">a<\/em> units above and below the center<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Rectangle<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Use <em data-effect=\"italics\">a<\/em> units left\/right of center<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">b<\/em> units above\/ below the center<\/td><td data-valign=\"middle\" data-align=\"center\">Use <em data-effect=\"italics\">a<\/em> units above\/below the center<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">b<\/em> units left\/right of center<\/td><\/tr><\/tbody><\/table><div data-type=\"example\" id=\"fs-id1163873659370\" class=\"textbox textbox--examples\"><div data-type=\"title\">How to Graph a Hyperbola with Center \\(\\left(h,k\\right)\\)<\/div><div data-type=\"exercise\" id=\"fs-id1163873628504\"><div data-type=\"problem\" id=\"fs-id1163873628506\"><p id=\"fs-id1163869142190\">Graph \\(\\frac{{\\left(x-1\\right)}^{2}}{9}-\\frac{{\\left(y-2\\right)}^{2}}{16}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873805530\"><span data-type=\"media\" id=\"fs-id1163873805533\" data-alt=\"Step 1 is to write the equation in standard form. Notice that the equation the quantity x minus 1 squared all divided by 9 end quantity minus the quantity y minus 2 squared all divided by 16 end quantity is equal to 1 is already in standard form.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the equation in standard form. Notice that the equation the quantity x minus 1 squared all divided by 9 end quantity minus the quantity y minus 2 squared all divided by 16 end quantity is equal to 1 is already in standard form.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873665542\" data-alt=\"Step 2 is to deteremine whether the transverse axis is horizonal or vertical. Since the x squared term is positive, the hyperbola opens left and right. The transverse axis is horizontal. The hyperbola opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to deteremine whether the transverse axis is horizonal or vertical. Since the x squared term is positive, the hyperbola opens left and right. The transverse axis is horizontal. The hyperbola opens left and right.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873769017\" data-alt=\"Step 3 is to find the center and a and b. h is equal to 1 and k is equal 2. a squared is equal to 9 and b squared is equal to 16. You can see tha x minus h is x minus 1, and that y minus k is y minus 2. So, the center is (1, 2) and a is equal to 3 and b is equal to 4.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to find the center and a and b. h is equal to 1 and k is equal 2. a squared is equal to 9 and b squared is equal to 16. You can see tha x minus h is x minus 1, and that y minus k is y minus 2. So, the center is (1, 2) and a is equal to 3 and b is equal to 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873952101\" data-alt=\"Step 4 is to sketch the rectangle centered at (h, k) using a and b. Mark the center (1, 2) on a coordinate plane. Sketch the rectangle that goes through the points 3 units to the left and right of the center and 4 units above and below the center.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to sketch the rectangle centered at (h, k) using a and b. Mark the center (1, 2) on a coordinate plane. Sketch the rectangle that goes through the points 3 units to the left and right of the center and 4 units above and below the center.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873863775\" data-alt=\"Step 5 is to sketch the asymptotes on the coordinate plane. They are the lines through the diagonals of the retcangle. Mark the vertices which lie on the rectangle 3 units to the left and right of the center. The vertices are (negative 2, 2) and (4, 2).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to sketch the asymptotes on the coordinate plane. They are the lines through the diagonals of the retcangle. Mark the vertices which lie on the rectangle 3 units to the left and right of the center. The vertices are (negative 2, 2) and (4, 2).\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873797607\" data-alt=\"Step 6 is to draw the branches of the hyperbola. Start the vertices, (negative 2, 2) and (4, 2) and use the asymptotes as a guide. The branches should open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to draw the branches of the hyperbola. Start the vertices, (negative 2, 2) and (4, 2) and use the asymptotes as a guide. The branches should open left and right.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873787440\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873787444\"><div data-type=\"problem\" id=\"fs-id1163873787446\"><p id=\"fs-id1163873607103\">Graph \\(\\frac{{\\left(x-3\\right)}^{2}}{25}-\\frac{{\\left(y-1\\right)}^{2}}{9}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873870786\"><span data-type=\"media\" id=\"fs-id1163873870789\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with an asymptote that passes through (negative 2, negative 2) and (8, 4) and an asymptote that passes through (negative 2, 4) and (8, negative 2), and branches that pass through the vertices (negative 2, 2) and (8, 2) and opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with an asymptote that passes through (negative 2, negative 2) and (8, 4) and an asymptote that passes through (negative 2, 4) and (8, negative 2), and branches that pass through the vertices (negative 2, 2) and (8, 2) and opens left and right.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873757662\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163870376223\"><div data-type=\"problem\" id=\"fs-id1163870376225\"><p id=\"fs-id1163870376227\">Graph \\(\\frac{{\\left(x-2\\right)}^{2}}{4}-\\frac{{\\left(y-2\\right)}^{2}}{9}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873765727\"><span data-type=\"media\" id=\"fs-id1163873629600\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (2, 2), an asymptote that passes through (0, negative 1) and (4, 5) and an asymptote that passes through (0, 5) and (4, negative 1), and branches that pass through the vertices (0, 2) and (4, 2) and opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_307_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (2, 2), an asymptote that passes through (0, negative 1) and (4, 5) and an asymptote that passes through (0, 5) and (4, negative 1), and branches that pass through the vertices (0, 2) and (4, 2) and opens left and right.\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163873659703\">We summarize the steps for easy reference.<\/p><div data-type=\"note\" id=\"fs-id1163873659707\" class=\"howto\"><div data-type=\"title\">Graph a hyperbola centered at \\(\\left(h,k\\right).\\)<\/div><ol id=\"fs-id1163873823602\" type=\"1\" class=\"stepwise\"><li>Write the equation in standard form.<\/li><li>Determine whether the transverse axis is horizontal or vertical.<\/li><li>Find the center and <em data-effect=\"italics\">a, b<\/em>.<\/li><li>Sketch the rectangle centered at \\(\\left(h,k\\right)\\) using <em data-effect=\"italics\">a, b<\/em>.<\/li><li>Sketch the asymptotes\u2014the lines through the diagonals of the rectangle. Mark the vertices.<\/li><li>Draw the two branches of the hyperbola.<\/li><\/ol><\/div><p id=\"fs-id1163873623710\">Be careful as you identify the center. The standard equation has \\(x-h\\) and \\(y-k\\) with the center as \\(\\left(h,k\\right).\\)<\/p><div data-type=\"example\" id=\"fs-id1163873796848\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873796850\"><div data-type=\"problem\" id=\"fs-id1163873620826\"><p id=\"fs-id1163873620828\">Graph \\(\\frac{{\\left(y+2\\right)}^{2}}{9}-\\frac{{\\left(x+1\\right)}^{2}}{4}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870292411\"><table id=\"fs-id1163873864599\" class=\"unnumbered unstyled\" summary=\"The quantity y plus 2 squared all divided by 9 end quantity minus the quantity x plus 1 all divided by 4 end quantity is equal to 1. Since the y squared term is positive, the hyperbola opens up and down. The expression y minus k is given by y minus negative 2 and the expression x minus h is given by x minus negative 1. Find the center, (h, k). Find a and b. The center is (negative 1, negative 2) and a is equal to 3 and b is equal to 2. Sketch the rectangle that goes through the points 3 units above and below the center and 2 units to the left and right of the center. Sketch the asymptotes, the lines through the diagonals of the rectangle. One asymptote passes through (negative 3, negative 5) and (1, 1) and the other passes through (negative 3, 1) and (1, negative 5) Mark the vertices. Graph the branches, which pass through the vertices (negative 3, negative 2) and (1, negative 2) and open up and down.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870345938\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since the \\({y}^{2}\\text{-}\\)term is positive, the hyperbola<span data-type=\"newline\"><br \/><\/span>opens up and down.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873703032\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the center, \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">Center: \\(\\left(-1,-2\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find <em data-effect=\"italics\">a, b<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(a=3\\)\\(b=2\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the rectangle that goes through the<span data-type=\"newline\"><br \/><\/span>points 3 units above and below the center and<span data-type=\"newline\"><br \/><\/span>2 units to the left\/right of the center.<span data-type=\"newline\"><br \/><\/span>Sketch the asymptotes\u2014the lines through the<span data-type=\"newline\"><br \/><\/span>diagonals of the rectangle.<span data-type=\"newline\"><br \/><\/span>Mark the vertices.<span data-type=\"newline\"><br \/><\/span>Graph the branches.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869585742\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873732322\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163870254062\"><div data-type=\"problem\" id=\"fs-id1163870254064\"><p id=\"fs-id1163870254067\">Graph \\(\\frac{{\\left(y+3\\right)}^{2}}{16}-\\frac{{\\left(x+2\\right)}^{2}}{9}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873861041\"><span data-type=\"media\" id=\"fs-id1163873861044\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 3), an asymptote that passes through (negative 5, negative 7) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, 7), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 7) and opens up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 3), an asymptote that passes through (negative 5, negative 7) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, 7), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 7) and opens up and down.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873558405\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873789429\"><div data-type=\"problem\" id=\"fs-id1163873789431\"><p id=\"fs-id1163873789433\">Graph \\(\\frac{{\\left(y+2\\right)}^{2}}{9}-\\frac{{\\left(x+2\\right)}^{2}}{9}=1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873884826\"><span data-type=\"media\" id=\"fs-id1163873605470\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 2), an asymptote that passes through (negative 5, negative 5) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, negative 5), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 5) and opens up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_309_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 2), an asymptote that passes through (negative 5, negative 5) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, negative 5), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 5) and opens up and down.\" \/><\/span><\/div><\/div><\/div><p id=\"fs-id1163867181069\">Again, sometimes we have to put the equation in standard form as our first step.<\/p><div data-type=\"example\" id=\"fs-id1163873743007\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873743009\"><div data-type=\"problem\" id=\"fs-id1163873743011\"><p id=\"fs-id1163873743013\">Write the equation in standard form and graph \\(4{x}^{2}-9{y}^{2}-24x-36y-36=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870548674\"><table id=\"fs-id1163870548677\" class=\"unnumbered unstyled can-break\" summary=\"4 x squared minus 9 y squared minus 24 x minus 36 y minus 36 is equal to 0. To get to standard form, complete the squares. 4 times the quantity x squared minus 6 x blank end quantity minus 9 times the quantity y squared plus 4 y blank) end quantity is equal to 36. 4 times the quantity x squared minus 6 x plus 9 end quantity minus 9 times the quantity y squared plus 4 y plus 4 end quantity is equal to 36 plus 36 minus 36. 4 times the quantity x minus 3 squared minus 9 times the quantity y plus 2 squared is equal to 36. Divide each term by 36 to get the constant to be 1. 4 times the quantity x minus 3 squared all divided by 36 minus 9 times the quantity y plus 2 squared all divided by 36 is equal to 36 divided by 36. The result is the quantity x minus 3 squared all divided by 36 minus the quantity y plus 2 squared all divided by 4 is equal to 1. Since the x squared term is positive, the hyperbola opens left and right. Find the center, (h, k). The center is (3, negative 2). Find a and b. a is equal to 3 and b is equal to 4. Sketch the rectangle that goes through the points 3 units to the left and right of the center and 2 units above and below the center. Sketch the asymptotes, the lines through the diagonals of the rectangle. One asymptote passes through (0, 0) and (6, negative 4) and the other passes through (0, negative 4) and (6, 0). Mark the vertices at (0, negative 2) and (6, negative 2). Graph the branches, making sure that they pass through the vertices.\" data-label=\"\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873791834\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">To get to standard form, complete the squares.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873648496\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870463090\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873764283\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Divide each term by 36 to get the constant to be 1.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163867240277\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\"><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873792475\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Since the \\({x}^{2}\\text{-}\\)term is positive, the hyperbola<span data-type=\"newline\"><br \/><\/span>opens left and right.<\/td><td data-valign=\"top\" data-align=\"center\"><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find the center, \\(\\left(h,k\\right).\\)<\/td><td data-valign=\"top\" data-align=\"center\">Center: \\(\\left(3,-2\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Find <em data-effect=\"italics\">a, b<\/em>.<\/td><td data-valign=\"top\" data-align=\"center\">\\(\\begin{array}{c}a=3\\hfill \\\\ b=4\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"left\">Sketch the rectangle that goes through the<span data-type=\"newline\"><br \/><\/span>points 3 units to the left\/right of the center<span data-type=\"newline\"><br \/><\/span>and 2 units above and below the center.<span data-type=\"newline\"><br \/><\/span>Sketch the asymptotes\u2014the lines through the<span data-type=\"newline\"><br \/><\/span>diagonals of the rectangle.<span data-type=\"newline\"><br \/><\/span>Mark the vertices.<span data-type=\"newline\"><br \/><\/span>Graph the branches.<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873644340\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163870291828\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873626188\"><div data-type=\"problem\" id=\"fs-id1163873626190\"><p id=\"fs-id1163873626192\"><span class=\"token\">\u24d0<\/span> Write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph \\(9{x}^{2}-16{y}^{2}+18x+64y-199=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870276462\"><p id=\"fs-id1163873799558\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(x+1\\right)}^{2}}{16}-\\frac{{\\left(y-2\\right)}^{2}}{9}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873784873\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (negative 1, 2), an asymptote that passes through (negative 5, 5) and (3, negative 1) and an asymptote that passes through (3, 5) and (negative 5, negative 1), and branches that pass through the vertices (negative 5, 2) and (3, 2) and opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (negative 1, 2), an asymptote that passes through (negative 5, 5) and (3, negative 1) and an asymptote that passes through (3, 5) and (negative 5, negative 1), and branches that pass through the vertices (negative 5, 2) and (3, 2) and opens left and right.\" \/><\/span><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163869435918\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163869435921\"><div data-type=\"problem\" id=\"fs-id1163869435923\"><p id=\"fs-id1163869435925\"><span class=\"token\">\u24d0<\/span> Write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph \\(16{x}^{2}-25{y}^{2}+96x-50y-281=0.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873872203\"><p id=\"fs-id1163873872205\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(x+3\\right)}^{2}}{25}-\\frac{{\\left(y+1\\right)}^{2}}{16}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873803299\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (negative 3, negative 1), an asymptote that passes through (negative 8, negative 5) and (2, 3) and an asymptote that passes through (negative 8, 3) and (2, negative 5), and branches that pass through the vertices (negative 8, negative 1) and (2, negative 1) and opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_311_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (negative 3, negative 1), an asymptote that passes through (negative 8, negative 5) and (2, 3) and an asymptote that passes through (negative 8, 3) and (2, negative 5), and branches that pass through the vertices (negative 8, negative 1) and (2, negative 1) and opens left and right.\" \/><\/span><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873662594\"><h3 data-type=\"title\">Identify Conic Sections by their Equations<\/h3><p id=\"fs-id1163873662599\">Now that we have completed our study of the conic sections, we will take a look at the different equations and recognize some ways to identify a conic by its equation. When we are given an equation to graph, it is helpful to identify the conic so we know what next steps to take.<\/p><p id=\"fs-id1163870414147\">To identify a conic from its equation, it is easier if we put the variable terms on one side of the equation and the constants on the other.<\/p><table id=\"fs-id1163870414151\" summary=\"This table has three columns and five rows. The first row is a header row and it labels each column, &#x201c;Conic,&#x201d; &#x201c;Characteristics of x squared and y squared terms,&#x201d; and &#x201c;Example.&#x201d; The first column is a header column and it labels each row &#x201c;Parabola,&#x201d; &#x201c;Circle,&#x201d; &#x201c;Ellipse,&#x201d;, and &#x201c;Hyperbola.&#x201d; In row two, the Parabola is described as having either x squared or y squared and only one variable squared and the example is x is equal to 3 y squared minus 2 y plus 1. In row three, the Circle is described as having x squared and y squared terms with the same coefficients and the example is x squared plus y squared is equal to 49. In row four, the Ellipse is described as having x squared and y squared terms that have the same sign and different coefficients and the example is 4 x squared plus 25 y squared is equal to 100. In row five, the Hyperbola is described as having x squared and y squared terms that have different signs and different coefficients and the example is 25 y squared minus 4 x squared is equal to 100.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"left\">Conic<\/th><th data-valign=\"middle\" data-align=\"left\">Characteristics of \\({x}^{2}\\text{-}\\) and \\({y}^{2}\\text{-}\\) terms<\/th><th data-valign=\"middle\" data-align=\"left\">Example<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Parabola<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">Either \\({x}^{2}\\) OR \\({y}^{2}.\\) Only one variable is squared.<\/td><td data-valign=\"middle\" data-align=\"left\">\\(x=3{y}^{2}-2y+1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Circle<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">\\({x}^{2}\\text{-}\\) and \\({y}^{2}\\text{-}\\) terms have the same coefficients<\/td><td data-valign=\"middle\" data-align=\"left\">\\({x}^{2}+{y}^{2}=49\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Ellipse<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">\\({x}^{2}\\text{-}\\) and \\({y}^{2}\\text{-}\\) terms have the <strong data-effect=\"bold\">same<\/strong> sign, different coefficients<\/td><td data-valign=\"middle\" data-align=\"left\">\\(4{x}^{2}+25{y}^{2}=100\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Hyperbola<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">\\({x}^{2}\\text{-}\\) and \\({y}^{2}\\text{-}\\) terms have <strong data-effect=\"bold\">different<\/strong> signs, different coefficients<\/td><td data-valign=\"middle\" data-align=\"left\">\\(25{y}^{2}-4{x}^{2}=100\\)<\/td><\/tr><\/tbody><\/table><div data-type=\"example\" id=\"fs-id1163873621465\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1163873621467\"><div data-type=\"problem\" id=\"fs-id1163873621469\"><p id=\"fs-id1163873621471\">Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.<\/p><p id=\"fs-id1163873655251\"><span class=\"token\">\u24d0<\/span>\\(9{x}^{2}+4{y}^{2}+56y+160=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span>\\(9{x}^{2}-16{y}^{2}+18x+64y-199=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span>\\({x}^{2}+{y}^{2}-6x-8y=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span>\\(y=-2{x}^{2}-4x-5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870346275\"><p id=\"fs-id1163873599316\"><span class=\"token\">\u24d0<\/span><span data-type=\"newline\"><br \/><\/span>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{6em}{0ex}}9{x}^{2}+4{y}^{2}+56y+160=0\\hfill \\\\ \\begin{array}{c}\\text{The}\\phantom{\\rule{0.2em}{0ex}}{x}^{2}\\text{-}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{y}^{2}\\text{-terms have the same sign and}\\hfill \\\\ \\text{different coefficients.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{6em}{0ex}}\\text{Ellipse}\\hfill \\end{array}\\)<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}9{x}^{2}-16{y}^{2}+18x+64y-199=0\\hfill \\\\ \\begin{array}{c}\\text{The}\\phantom{\\rule{0.2em}{0ex}}{x}^{2}\\text{-}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{y}^{2}\\text{-terms have different signs and}\\hfill \\\\ \\text{different coefficients.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{4em}{0ex}}\\text{Hyperbola}\\hfill \\end{array}\\)<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/><\/span>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{5.5em}{0ex}}{x}^{2}+{y}^{2}-6x-8y=0\\hfill \\\\ \\text{The}\\phantom{\\rule{0.2em}{0ex}}{x}^{2}\\text{-}\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{y}^{2}\\text{-terms have the same coefficients.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{5.5em}{0ex}}\\text{Circle}\\hfill \\end{array}\\)<span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><span data-type=\"newline\"><br \/><\/span>\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{12em}{0ex}}y=-2{x}^{2}-4x-5\\hfill \\\\ \\text{Only one variable,}\\phantom{\\rule{0.2em}{0ex}}x,\\phantom{\\rule{0.2em}{0ex}}\\text{is squared.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{13em}{0ex}}\\text{Parabola}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163866895479\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873744525\"><div data-type=\"problem\" id=\"fs-id1163873744527\"><p id=\"fs-id1163873744529\">Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.<\/p><p id=\"fs-id1163873744532\"><span class=\"token\">\u24d0<\/span>\\({x}^{2}+{y}^{2}-8x-6y=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span>\\(4{x}^{2}+25{y}^{2}=100\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span>\\(y=6{x}^{2}+2x-1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span>\\(16{y}^{2}-9{x}^{2}=144\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873650679\"><p id=\"fs-id1163873650681\"><span class=\"token\">\u24d0<\/span> circle <span class=\"token\">\u24d1<\/span> ellipse <span class=\"token\">\u24d2<\/span> parabola <span class=\"token\">\u24d3<\/span> hyperbola<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163870407092\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1163873861534\"><div data-type=\"problem\" id=\"fs-id1163873861536\"><p id=\"fs-id1163873861538\">Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.<\/p><p id=\"fs-id1163873861542\"><span class=\"token\">\u24d0<\/span>\\(16{x}^{2}+9{y}^{2}=144\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span>\\(y=2{x}^{2}+4x+6\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span>\\({x}^{2}+{y}^{2}+2x+6y+9=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span>\\(4{x}^{2}-16{y}^{2}=64\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873667027\"><p id=\"fs-id1163873667029\"><span class=\"token\">\u24d0<\/span> ellipse <span class=\"token\">\u24d1<\/span> parabola <span class=\"token\">\u24d2<\/span> circle <span class=\"token\">\u24d3<\/span> hyperbola<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1163873606913\" class=\"media-2\"><p id=\"fs-id1163873606917\">Access these online resources for additional instructions and practice with hyperbolas.<\/p><ul id=\"fs-id1163873606921\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37ghyperborig\">Graph a Hyperbola with Center at the Origin<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37ghyperbnorig\">Graph a Hyperbola with Center not at the Origin<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37ghyperbgen\">Graph a Hyperbola in General Form<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37conicsgen\">Identifying Conic Sections in General Form<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163873870675\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1163873793083\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Hyperbola:<\/strong> A <strong data-effect=\"bold\">hyperbola<\/strong> is all points in a plane where the difference of their distances from two fixed points is constant.<span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1163869200311\" data-alt=\"The figure shows a double napped right circular cone sliced by a plane that is parallel to the vertical axis of the cone forming a hyperbola. The figure is labeled &#x2018;hyperbola&#x2019;.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_011_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a double napped right circular cone sliced by a plane that is parallel to the vertical axis of the cone forming a hyperbola. The figure is labeled &#x2018;hyperbola&#x2019;.\" \/><\/span><span data-type=\"newline\"><br \/><\/span> Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the hyperbola.<span data-type=\"newline\"><br \/><\/span> The line through the foci, is called the <strong data-effect=\"bold\">transverse axis<\/strong>.<span data-type=\"newline\"><br \/><\/span> The two points where the transverse axis intersects the hyperbola are each a <strong data-effect=\"bold\">vertex<\/strong> of the hyperbola.<span data-type=\"newline\"><br \/><\/span> The midpoint of the segment joining the foci is called the <strong data-effect=\"bold\">center<\/strong> of the hyperbola.<span data-type=\"newline\"><br \/><\/span> The line perpendicular to the transverse axis that passes through the center is called the <strong data-effect=\"bold\">conjugate axis<\/strong>.<span data-type=\"newline\"><br \/><\/span> Each piece of the graph is called a <strong data-effect=\"bold\">branch<\/strong> of the hyperbola.<span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1163873652586\" data-alt=\"The figure shows two graphs of a hyperbola. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci are shown with points that lie on the transverse axis, which is the x-axis. The branches pass through the vertices and open left and right. The y-axis is the conjugate axis. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci lie are shown with points that lie on the transverse axis, which is the y-axis. The branches pass through the vertices and open up and down. The x-axis is the conjugate axis.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows two graphs of a hyperbola. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci are shown with points that lie on the transverse axis, which is the x-axis. The branches pass through the vertices and open left and right. The y-axis is the conjugate axis. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci lie are shown with points that lie on the transverse axis, which is the y-axis. The branches pass through the vertices and open up and down. The x-axis is the conjugate axis.\" \/><\/span><span data-type=\"newline\"><br \/><\/span> <table id=\"fs-id1163873677570\" summary=\"The table has three columns and eight rows. The first row is a title row and is labeled &#x2018;Standard Forms of the Equation a Hyperbola with Center (0, 0). The second row is a header row with the first column labeled the quantity x squared divided by a squared end quantity minus the quantity y squared divided by b squared is equal to 1 and the second column labeled the quantity y squared divided by a squared end quantity minus the quantity x squared divided by b squared is equal to 1. The rows are labeled &#x2018;Orientation&#x2019;, &#x2018;Vertices&#x2019;, &#x2018;x-intercepts&#x2019;, &#x2018;y-intercepts&#x2019;, &#x2018;Rectangle&#x2019;, and &#x2018;Asymptotes. In row three, the &#x2018;Orientations&#x2019; are &#x2018;transverse axis on the x-axis; opens left and right&#x2019; and &#x2018;transverse axis on the y-axis; opens up and down&#x2019;. In row four, the &#x2018;Vertices&#x2019; are (negative a, 0) and (a, 0) and (0, negative a) and (0, a). In row five, the x-intercepts are (negative a, 0) and (a, 0) and &#x2018;none). In row six, the y-intercepts are &#x2018;none&#x2019; and (0, negative a) and (0, a). In row seven, the rectangle uses (plus or minus a, 0) and (0, plus or minus) and uses (0, plus or minus a) and (plus or minus b, 0). In row eight, the asymptotes are y is equal to b divided by a times x, y is equal to negative b divided by a times x and y is equal to a divided by b times x and y is equal to negative a divided by b times x.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Standard Forms of the Equation a Hyperbola with Center \\(\\left(0,0\\right)\\)<\/th><\/tr><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"center\"><\/th><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{x}^{2}}{{a}^{2}}-\\frac{{y}^{2}}{{b}^{2}}=1\\)<\/th><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{y}^{2}}{{a}^{2}}-\\frac{{x}^{2}}{{b}^{2}}=1\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Orientation<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Transverse axis on the <em data-effect=\"italics\">x<\/em>-axis.<span data-type=\"newline\"><br \/><\/span>Opens left and right<\/td><td data-valign=\"middle\" data-align=\"center\">Transverse axis on the <em data-effect=\"italics\">y<\/em>-axis.<span data-type=\"newline\"><br \/><\/span>Opens up and down<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Vertices<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(\\text{\u2212}a,0\\right),\\)\\(\\left(a,0\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,\\text{\u2212}a\\right),\\)\\(\\left(0,a\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(\\text{\u2212}a,0\\right),\\)\\(\\left(a,0\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"center\">none<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">none<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(0,\\text{\u2212}a\\right)\\), \\(\\left(0,a\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Rectangle<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Use \\(\\left(\\text{\u00b1}a,0\\right)\\) \\(\\left(0,\\text{\u00b1}b\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"center\">Use \\(\\left(0,\\text{\u00b1}a\\right)\\) \\(\\left(\\text{\u00b1}b,0\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">asymptotes<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(y=\\frac{b}{a}x,\\)\\(y=-\\frac{b}{a}x\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(y=\\frac{a}{b}x,\\)\\(y=-\\frac{a}{b}x\\)<\/td><\/tr><\/tbody><\/table><\/li><li><strong data-effect=\"bold\">How to graph a hyperbola centered at \\(\\left(0,0\\right).\\)<\/strong><ol id=\"fs-id1163873898656\" type=\"1\" class=\"stepwise\"><li>Write the equation in standard form.<\/li><li>Determine whether the transverse axis is horizontal or vertical.<\/li><li>Find the vertices.<\/li><li>Sketch the rectangle centered at the origin intersecting one axis at \\(\\text{\u00b1}a\\) and the other at \\(\\text{\u00b1}b.\\)<\/li><li>Sketch the asymptotes\u2014the lines through the diagonals of the rectangle.<\/li><li>Draw the two branches of the hyperbola.<\/li><\/ol><span data-type=\"newline\"><br \/><\/span><table id=\"fs-id1163873632917\" summary=\"The table has three columns and six rows. The first row is a title row and is labeled &#x2018;Standard Forms of the Equation a Hyperbola with Center (h, k). The second row is a header row with the first column labeled the quantity x minus h squared all divided by a squared end quantity minus the quantity y minus k squared all divided by b squared end quantity is equal to 1 and the second column labeled the quantity y minus k squared all divided by a squared end quantity minus the quantity x minus h squared all divided by b squared end quantity is equal to 1. The rows are labeled &#x2018;Orientation&#x2019;, &#x2018;Center&#x2019;, &#x2018;Vertices&#x2019;, and &#x2018;Rectangle&#x2019;. In row three, the &#x2018;Orientations&#x2019; are &#x2018;transverse axis on the x-axis; opens left and right&#x2019; and &#x2018;transverse axis on the y-axis; opens up and down&#x2019;. In row four, the &#x2018;Centers&#x2019; are both (h, k). In row five, the &#x2018;Vertices&#x2019; are a units to the left and right of the center and a units above and below the center. In row six, the &#x2018;Rectangles&#x2019; are formed by moving a units left or right of the center and b units above or below the center, and by using a units above or below the center and b units left or right of the center.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Standard Forms of the Equation a Hyperbola with Center \\(\\left(h,k\\right)\\)<\/th><\/tr><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"center\"><\/th><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{\\left(x-h\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(y-k\\right)}^{2}}{{b}^{2}}=1\\)<\/th><th data-valign=\"middle\" data-align=\"center\">\\(\\frac{{\\left(y-k\\right)}^{2}}{{a}^{2}}-\\frac{{\\left(x-h\\right)}^{2}}{{b}^{2}}=1\\)<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Orientation<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Transverse axis is horizontal.<span data-type=\"newline\"><br \/><\/span>Opens left and right<\/td><td data-valign=\"middle\" data-align=\"center\">Transverse axis is vertical.<span data-type=\"newline\"><br \/><\/span>Opens up and down<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Center<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(h,k\\right)\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\left(h,k\\right)\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Vertices<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><em data-effect=\"italics\">a<\/em> units to the left and right of the center<\/td><td data-valign=\"middle\" data-align=\"center\"><em data-effect=\"italics\">a<\/em> units above and below the center<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Rectangle<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">Use <em data-effect=\"italics\">a<\/em> units left\/right of center<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">b<\/em> units above\/below the center<\/td><td data-valign=\"middle\" data-align=\"center\">Use <em data-effect=\"italics\">a<\/em> units above\/below the center<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">b<\/em> units left\/right of center<\/td><\/tr><\/tbody><\/table><\/li><li><strong data-effect=\"bold\">How to graph a hyperbola centered at \\(\\left(h,k\\right).\\)<\/strong><ol id=\"fs-id1163873664094\" type=\"1\" class=\"stepwise\"><li>Write the equation in standard form.<\/li><li>Determine whether the transverse axis is horizontal or vertical.<\/li><li>Find the center and \\(a,b.\\)<\/li><li>Sketch the rectangle centered at \\(\\left(h,k\\right)\\) using \\(a,b.\\)<\/li><li>Sketch the asymptotes\u2014the lines through the diagonals of the rectangle. Mark the vertices.<\/li><li>Draw the two branches of the hyperbola.<\/li><\/ol><span data-type=\"newline\"><br \/><\/span><table id=\"fs-id1163873558902\" summary=\"This table has three columns and five rows. The first row is a header row and it labels each column, &#x201c;Conic,&#x201d; &#x201c;Characteristics of x squared and y squared terms,&#x201d; and &#x201c;Example.&#x201d; The first column is a header column and it labels each row &#x201c;Parabola,&#x201d; &#x201c;Circle,&#x201d; &#x201c;Ellipse,&#x201d;, and &#x201c;Hyperbola.&#x201d; In row two, the Parabola is described as having either x squared or y squared and only one variable squared and the example is x is equal to 3 y squared minus 2 y plus 1. In row three, the Circle is described as having x squared and y squared terms with the same coefficients and the example is x squared plus y squared is equal to 49. In row four, the Ellipse is described as having x squared and y squared terms that have the same sign and different coefficients and the example is 4 x squared plus 25 y squared is equal to 100. In row five, the Hyperbola is described as having x squared and y squared terms that have different signs and different coefficients and the example is 25 y squared minus 4 x squared is equal to 100.\" class=\"unnumbered\" data-label=\"\"><thead><tr valign=\"top\"><th data-valign=\"middle\" data-align=\"left\">Conic<\/th><th data-valign=\"middle\" data-align=\"left\">Characteristics of \\({x}^{2}\\text{-}\\) and \\({y}^{2}\\text{-}\\) terms<\/th><th data-valign=\"middle\" data-align=\"left\">Example<\/th><\/tr><\/thead><tbody><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Parabola<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">Either \\({x}^{2}\\) OR \\({y}^{2}.\\) Only one variable is squared.<\/td><td data-valign=\"middle\" data-align=\"left\">\\(x=3{y}^{2}-2y+1\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Circle<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">\\({x}^{2}\\text{-}\\) and \\({y}^{2}\\text{-}\\) terms have the same coefficients<\/td><td data-valign=\"middle\" data-align=\"left\">\\({x}^{2}+{y}^{2}=49\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Ellipse<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">\\({x}^{2}\\text{-}\\) and \\({y}^{2}\\text{-}\\) terms have the <strong data-effect=\"bold\">same<\/strong> sign, different coefficients<\/td><td data-valign=\"middle\" data-align=\"left\">\\(4{x}^{2}+25{y}^{2}=100\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Hyperbola<\/strong><\/td><td data-valign=\"middle\" data-align=\"left\">\\({x}^{2}\\text{-}\\) and \\({y}^{2}\\text{-}\\) terms have <strong data-effect=\"bold\">different<\/strong> signs, different coefficients<\/td><td data-valign=\"middle\" data-align=\"left\">\\(25{y}^{2}-4{x}^{2}=100\\)<\/td><\/tr><\/tbody><\/table><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163873814241\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1163873782017\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1163873782024\"><strong data-effect=\"bold\">Graph a Hyperbola with Center at \\(\\left(0,0\\right)\\)<\/strong><\/p><p id=\"fs-id1163874053963\">In the following exercises, graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163874053966\"><div data-type=\"problem\" id=\"fs-id1163873513529\"><p id=\"fs-id1163873513531\">\\(\\frac{{x}^{2}}{9}-\\frac{{y}^{2}}{4}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870368930\"><span data-type=\"media\" id=\"fs-id1163870368933\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus two-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus two-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163874047591\"><div data-type=\"problem\" id=\"fs-id1163874047593\"><p id=\"fs-id1163874047595\">\\(\\frac{{x}^{2}}{25}-\\frac{{y}^{2}}{9}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873657989\"><div data-type=\"problem\" id=\"fs-id1163873657991\"><p id=\"fs-id1163873657993\">\\(\\frac{{x}^{2}}{16}-\\frac{{y}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163874047544\"><span data-type=\"media\" id=\"fs-id1163874047547\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-fourths times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-fourths times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873850658\"><div data-type=\"problem\" id=\"fs-id1163870549374\"><p id=\"fs-id1163870549376\">\\(\\frac{{x}^{2}}{9}-\\frac{{y}^{2}}{36}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163874019191\"><div data-type=\"problem\" id=\"fs-id1163874019193\"><p id=\"fs-id1163873651595\">\\(\\frac{{y}^{2}}{25}-\\frac{{x}^{2}}{4}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873725271\"><span data-type=\"media\" id=\"fs-id1163873678231\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_316_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873627622\"><div data-type=\"problem\" id=\"fs-id1163873627625\"><p id=\"fs-id1163873627627\">\\(\\frac{{y}^{2}}{36}-\\frac{{x}^{2}}{16}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163869138006\"><div data-type=\"problem\" id=\"fs-id1163869138008\"><p id=\"fs-id1163869138010\">\\(16{y}^{2}-9{x}^{2}=144\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163869142597\"><span data-type=\"media\" id=\"fs-id1163867248314\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-fourths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_318_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-fourths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870384297\"><div data-type=\"problem\" id=\"fs-id1163870384299\"><p id=\"fs-id1163870384301\">\\(25{y}^{2}-9{x}^{2}=225\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873787573\"><div data-type=\"problem\" id=\"fs-id1163873787575\"><p id=\"fs-id1163870218685\">\\(4{y}^{2}-9{x}^{2}=36\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873506044\"><span data-type=\"media\" id=\"fs-id1163873506047\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-halves times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_320_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-halves times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163869200126\"><div data-type=\"problem\" id=\"fs-id1163869200128\"><p id=\"fs-id1163869200130\">\\(16{y}^{2}-25{x}^{2}=400\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873793246\"><div data-type=\"problem\" id=\"fs-id1163873793248\"><p id=\"fs-id1163873793251\">\\(4{x}^{2}-16{y}^{2}=64\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873802105\"><span data-type=\"media\" id=\"fs-id1163873702446\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_322_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873814481\"><div data-type=\"problem\" id=\"fs-id1163873814483\"><p id=\"fs-id1163873814485\">\\(9{x}^{2}-4{y}^{2}=36\\)<\/p><\/div><\/div><p id=\"fs-id1163869406009\"><strong data-effect=\"bold\">Graph a Hyperbola with Center at \\(\\left(h,k\\right)\\)<\/strong><\/p><p id=\"fs-id1163873898758\">In the following exercises, graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163873898761\"><div data-type=\"problem\" id=\"fs-id1163873627636\"><p id=\"fs-id1163873627638\">\\(\\frac{{\\left(x-1\\right)}^{2}}{16}-\\frac{{\\left(y-3\\right)}^{2}}{4}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163869164198\"><span data-type=\"media\" id=\"fs-id1163869164201\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_324_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873655036\"><div data-type=\"problem\" id=\"fs-id1163873655038\"><p id=\"fs-id1163873633345\">\\(\\frac{{\\left(x-2\\right)}^{2}}{4}-\\frac{{\\left(y-3\\right)}^{2}}{16}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873607551\"><div data-type=\"problem\" id=\"fs-id1163873607553\"><p id=\"fs-id1163873607555\">\\(\\frac{{\\left(y-4\\right)}^{2}}{9}-\\frac{{\\left(x-2\\right)}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873599765\"><span data-type=\"media\" id=\"fs-id1163873599768\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_326_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163874032597\"><div data-type=\"problem\" id=\"fs-id1163874032599\"><p id=\"fs-id1163874032602\">\\(\\frac{{\\left(y-1\\right)}^{2}}{25}-\\frac{{\\left(x-4\\right)}^{2}}{16}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870462018\"><div data-type=\"problem\" id=\"fs-id1163870462020\"><p id=\"fs-id1163870462022\">\\(\\frac{{\\left(y+4\\right)}^{2}}{25}-\\frac{{\\left(x+1\\right)}^{2}}{36}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163867240292\"><span data-type=\"media\" id=\"fs-id1163867240295\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, negative 4) an asymptote that passes through (negative 7, 1) and (5, negative 9) and an asymptote that passes through (5, 1) and (negative 7, negative 9), and branches that pass through the vertices (1, 1) and (1, negative 9) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_328_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, negative 4) an asymptote that passes through (negative 7, 1) and (5, negative 9) and an asymptote that passes through (5, 1) and (negative 7, negative 9), and branches that pass through the vertices (1, 1) and (1, negative 9) and open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873768786\"><div data-type=\"problem\" id=\"fs-id1163873768789\"><p id=\"fs-id1163873768791\">\\(\\frac{{\\left(y+1\\right)}^{2}}{16}-\\frac{{\\left(x+1\\right)}^{2}}{4}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873639294\"><div data-type=\"problem\" id=\"fs-id1163873639296\"><p id=\"fs-id1163869408283\">\\(\\frac{{\\left(y-4\\right)}^{2}}{16}-\\frac{{\\left(x+1\\right)}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873664555\"><span data-type=\"media\" id=\"fs-id1163873731499\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 1, 4) an asymptote that passes through (4, 8) and (negative 6, 0) and an asymptote that passes through (negative 6, 8) and (4, 0), and branches that pass through the vertices (negative 1, 0) and (negative 1, 8) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_330_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 1, 4) an asymptote that passes through (4, 8) and (negative 6, 0) and an asymptote that passes through (negative 6, 8) and (4, 0), and branches that pass through the vertices (negative 1, 0) and (negative 1, 8) and open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873753634\"><div data-type=\"problem\" id=\"fs-id1163873753636\"><p id=\"fs-id1163873753638\">\\(\\frac{{\\left(y+3\\right)}^{2}}{16}-\\frac{{\\left(x-3\\right)}^{2}}{36}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870489128\"><div data-type=\"problem\" id=\"fs-id1163870489131\"><p id=\"fs-id1163870489133\">\\(\\frac{{\\left(x-3\\right)}^{2}}{25}-\\frac{{\\left(y+2\\right)}^{2}}{9}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873664272\"><span data-type=\"media\" id=\"fs-id1163873664275\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (3, negative 2) an asymptote that passes through (8, 1) and (negative 2, negative 5) and an asymptote that passes through (negative 2, negative 1) and (8, negative 5), and branches that pass through the vertices (negative 2, negative 2) and (8, negative 2) and opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_332_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (3, negative 2) an asymptote that passes through (8, 1) and (negative 2, negative 5) and an asymptote that passes through (negative 2, negative 1) and (8, negative 5), and branches that pass through the vertices (negative 2, negative 2) and (8, negative 2) and opens left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873645239\"><div data-type=\"problem\" id=\"fs-id1163873645242\"><p id=\"fs-id1163873645244\">\\(\\frac{{\\left(x+2\\right)}^{2}}{4}-\\frac{{\\left(y-1\\right)}^{2}}{9}=1\\)<\/p><\/div><\/div><p id=\"fs-id1163873809201\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163870256623\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870256625\"><p id=\"fs-id1163870256627\">\\(9{x}^{2}-4{y}^{2}-18x+8y-31=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873865095\"><p id=\"fs-id1163873865097\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(x-1\\right)}^{2}}{4}-\\frac{{\\left(y-1\\right)}^{2}}{9}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163873783464\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 1) an asymptote that passes through (3, 4) and (negative 1, negative 2) and an asymptote that passes through (negative 1, 4) and (3, negative 2), and branches that pass through the vertices (negative 1, 1) and (3, 1) and opens left and right.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_334_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 1) an asymptote that passes through (3, 4) and (negative 1, negative 2) and an asymptote that passes through (negative 1, 4) and (3, negative 2), and branches that pass through the vertices (negative 1, 1) and (3, 1) and opens left and right.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873799258\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873799260\"><p id=\"fs-id1163873799262\">\\(16{x}^{2}-4{y}^{2}+64x-24y-36=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873866311\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873866313\"><p id=\"fs-id1163873866315\">\\({y}^{2}-{x}^{2}-4y+2x-6=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873633020\"><p id=\"fs-id1163873633022\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(y-2\\right)}^{2}}{9}-\\frac{{\\left(x-1\\right)}^{2}}{9}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163870407168\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 2) an asymptote that passes through (4, 5) and (negative 2, negative 1) and an asymptote that passes through (negative 2, 5) and (4, negative 1), and branches that pass through the vertices (1, 5) and (1, negative 1) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_336_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 2) an asymptote that passes through (4, 5) and (negative 2, negative 1) and an asymptote that passes through (negative 2, 5) and (4, negative 1), and branches that pass through the vertices (1, 5) and (1, negative 1) and open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873919210\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873906647\"><p id=\"fs-id1163873906649\">\\(4{y}^{2}-16{x}^{2}-24y+96x-172=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873819963\"><div data-type=\"problem\" id=\"fs-id1163873645897\"><p id=\"fs-id1163873645899\">\\(9{y}^{2}-{x}^{2}+18y-4x-4=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873648568\"><p id=\"fs-id1163873648570\"><span class=\"token\">\u24d0<\/span>\\(\\frac{{\\left(y+1\\right)}^{2}}{1}-\\frac{{\\left(x+2\\right)}^{2}}{9}=1\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p><span data-type=\"media\" id=\"fs-id1163870487208\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 2, negative 1) an asymptote that passes through (1, 0) and (negative 5, negative 2) and an asymptote that passes through (3, 0) and (1, negative 2), and branches that pass through the vertices (negative 2, 0) and (negative 2, negative 2) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_338_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 2, negative 1) an asymptote that passes through (1, 0) and (negative 5, negative 2) and an asymptote that passes through (3, 0) and (1, negative 2), and branches that pass through the vertices (negative 2, 0) and (negative 2, negative 2) and open up and down.\" \/><\/span><\/div><\/div><p id=\"fs-id1163870366282\"><strong data-effect=\"bold\">Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola<\/strong><\/p><p id=\"fs-id1163870516339\">In the following exercises, identify the type of graph.<\/p><div data-type=\"exercise\" id=\"fs-id1163870516342\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163870516344\"><p id=\"fs-id1163870516346\"><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d0<\/span>\\(x=\\text{\u2212}{y}^{2}-2y+3\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span>\\(9{y}^{2}-{x}^{2}+18y-4x-4=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span>\\(9{x}^{2}+25{y}^{2}=225\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span>\\({x}^{2}+{y}^{2}-4x+10y-7=0\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873796654\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1163873782194\"><p id=\"fs-id1163873782196\"><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d0<\/span>\\(x=-2{y}^{2}-12y-16\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span>\\({x}^{2}+{y}^{2}=9\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span>\\(16{x}^{2}-4{y}^{2}+64x-24y-36=0\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span>\\(16{x}^{2}+36{y}^{2}=576\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870619458\"><p id=\"fs-id1163870619460\"><span class=\"token\">\u24d0<\/span> parabola <span class=\"token\">\u24d1<\/span> circle <span class=\"token\">\u24d2<\/span> hyperbola <span class=\"token\">\u24d3<\/span> ellipse<\/p><\/div><\/div><p id=\"fs-id1163873648694\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p><p id=\"fs-id1163870259320\">In the following exercises, graph each equation.<\/p><div data-type=\"exercise\" id=\"fs-id1163870259323\"><div data-type=\"problem\" id=\"fs-id1163870259326\"><p id=\"fs-id1163870259328\">\\(\\frac{{\\left(y-3\\right)}^{2}}{9}-\\frac{{\\left(x+2\\right)}^{2}}{16}=1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873662392\"><div data-type=\"problem\" id=\"fs-id1163873869471\"><p id=\"fs-id1163873869473\">\\({x}^{2}+{y}^{2}-4x+10y-7=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873935276\"><span data-type=\"media\" id=\"fs-id1163873935279\" data-alt=\"The graph shows the x y coordinate plane with a circle whose center is (2, negative 5) and whose radius is 6 units.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_340_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x y coordinate plane with a circle whose center is (2, negative 5) and whose radius is 6 units.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873855097\"><div data-type=\"problem\" id=\"fs-id1163873855100\"><p id=\"fs-id1163873795847\">\\(y={\\left(x-1\\right)}^{2}+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870644270\"><div data-type=\"problem\" id=\"fs-id1163870644272\"><p id=\"fs-id1163870644274\">\\(\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{25}=1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873557315\"><span data-type=\"media\" id=\"fs-id1163873811845\" data-alt=\"The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 5) and co-vertices are (plus or minus 3, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_342_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 5) and co-vertices are (plus or minus 3, 0).\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870292668\"><div data-type=\"problem\" id=\"fs-id1163870292670\"><p id=\"fs-id1163870292672\">\\({\\left(x+2\\right)}^{2}+{\\left(y-5\\right)}^{2}=4\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873804950\"><div data-type=\"problem\" id=\"fs-id1163873804952\"><p id=\"fs-id1163873655564\">\\(9{x}^{2}-4{y}^{2}+54x+8y+41=0\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163870258892\"><span data-type=\"media\" id=\"fs-id1163870258895\" data-alt=\"The graph shows the x y coordinate plane with the center (1, 2) an asymptote that passes through (negative 2, 5) and (5, negative 1) and an asymptote that passes through (4, 5) and (2, 0), and branches that pass through the vertices (1, 5) and (negative 2, negative 1) and open up and down.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_344_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x y coordinate plane with the center (1, 2) an asymptote that passes through (negative 2, 5) and (5, negative 1) and an asymptote that passes through (4, 5) and (2, 0), and branches that pass through the vertices (1, 5) and (negative 2, negative 1) and open up and down.\" \/><\/span><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873748917\"><div data-type=\"problem\" id=\"fs-id1163873748920\"><p id=\"fs-id1163873748922\">\\(x=\\text{\u2212}{y}^{2}-2y+3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870549277\"><div data-type=\"problem\" id=\"fs-id1163870549279\"><p id=\"fs-id1163870549282\">\\(16{x}^{2}+9{y}^{2}=144\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873654059\"><span data-type=\"media\" id=\"fs-id1163870548126\" data-alt=\"The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 4) and co-vertices are (plus or minus 3, 0).\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_346_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 4) and co-vertices are (plus or minus 3, 0).\" \/><\/span><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1163874054189\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1163874054196\"><div data-type=\"problem\" id=\"fs-id1163873654001\"><p id=\"fs-id1163873654003\">In your own words, define a hyperbola and write the equation of a hyperbola centered at the origin in standard form. Draw a sketch of the hyperbola labeling the center, vertices, and asymptotes.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163870693901\"><div data-type=\"problem\" id=\"fs-id1163870693903\"><p id=\"fs-id1163870693905\">Explain in your own words how to create and use the rectangle that helps graph a hyperbola.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873641719\"><p id=\"fs-id1163873641721\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873641726\"><div data-type=\"problem\" id=\"fs-id1163873641728\"><p id=\"fs-id1163873641730\">Compare and contrast the graphs of the equations \\(\\frac{{x}^{2}}{4}-\\frac{{y}^{2}}{9}=1\\) and \\(\\frac{{y}^{2}}{9}-\\frac{{x}^{2}}{4}=1.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1163873716520\"><div data-type=\"problem\" id=\"fs-id1163873716522\"><p id=\"fs-id1163873716524\">Explain in your own words, how to distinguish the equation of an ellipse with the equation of a hyperbola.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1163873861987\"><p id=\"fs-id1163873861989\">Answers will vary.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163873861995\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1163874054311\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1163874054319\" data-alt=\"This table has four columns and four rows. The first row is a header and it labels each column, &#x201c;I can&#x2026;&#x201d;, &#x201c;Confidently,&#x201d; &#x201c;With some help,&#x201d; and &#x201c;No-I don&#x2019;t get it!&#x201d; In row 2, the I can was graph a hyperbola with center at (0, 0). In row 3, the I can was graph a hyperbola with a center at (h, k). In row 4, the I can was identify conic sections by their equations.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and four rows. The first row is a header and it labels each column, &#x201c;I can&#x2026;&#x201d;, &#x201c;Confidently,&#x201d; &#x201c;With some help,&#x201d; and &#x201c;No-I don&#x2019;t get it!&#x201d; In row 2, the I can was graph a hyperbola with center at (0, 0). In row 3, the I can was graph a hyperbola with a center at (h, k). In row 4, the I can was identify conic sections by their equations.\" \/><\/span><p id=\"fs-id1163873892733\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1163873673686\"><dt>hyperbola<\/dt><dd id=\"fs-id1163873673691\">A hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant.<\/dd><\/dl><\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Graph a hyperbola with center at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Graph a hyperbola with center at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Identify conic sections by their equations<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873928434\" class=\"be-prepared\">\n<p id=\"fs-id1163873605884\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1163873624750\" type=\"1\">\n<li>Solve: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-157d30cdf1bb4029cf2bcf811a5ec48a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/b9659e42-3afa-4449-81d9-a017c35de140#fs-id1167834228017\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Expand: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a642f02df02f10757178ae0d13026ac2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"65\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/0b9be1db-21c4-4bd0-8f8e-d809f6ff7c8c#fs-id1167836392219\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e82b05922e65169be331ebcf3751bd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"72\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/26e8f94c-1f76-46ec-8e6c-344f06971cf5#fs-id1167834408393\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163870348934\">\n<h3 data-type=\"title\">Graph a Hyperbola with Center at <strong data-effect=\"bold\">(0, 0)<\/strong><\/h3>\n<p id=\"fs-id1163873616698\">The last conic section we will look at is called a <span data-type=\"term\">hyperbola<\/span>. We will see that the equation of a hyperbola looks the same as the equation of an ellipse, except it is a difference rather than a sum. While the equations of an ellipse and a hyperbola are very similar, their graphs are very different.<\/p>\n<p id=\"fs-id1163870487157\">We define a <strong data-effect=\"bold\">hyperbola<\/strong> as all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the hyperbola.<\/p>\n<div data-type=\"note\" id=\"fs-id1163873631003\">\n<div data-type=\"title\">Hyperbola<\/div>\n<p id=\"fs-id1163873641928\">A <strong data-effect=\"bold\">hyperbola<\/strong> is all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the hyperbola.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873648722\" data-alt=\"The figure shows a double napped right circular cone sliced by a plane that is parallel to the vertical axis of the cone forming a hyperbola. The figure is labeled &#x2018;hyperbola&#x2019;.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a double napped right circular cone sliced by a plane that is parallel to the vertical axis of the cone forming a hyperbola. The figure is labeled &#x2018;hyperbola&#x2019;.\" \/><\/span><\/div>\n<p id=\"fs-id1163873644949\">The line through the foci, is called the <strong data-effect=\"bold\">transverse axis<\/strong>. The two points where the transverse axis intersects the hyperbola are each a <strong data-effect=\"bold\">vertex<\/strong> of the hyperbola. The midpoint of the segment joining the foci is called the <strong data-effect=\"bold\">center<\/strong> of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the <strong data-effect=\"bold\">conjugate axis<\/strong>. Each piece of the graph is called a <strong data-effect=\"bold\">branch<\/strong> of the hyperbola.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163870489434\" data-alt=\"The figure shows two graphs of a hyperbola. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci are shown with points that lie on the transverse axis, which is the x-axis. The branches pass through the vertices and open left and right. The y-axis is the conjugate axis. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci lie are shown with points that lie on the transverse axis, which is the y-axis. The branches pass through the vertices and open up and down. The x-axis is the conjugate axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows two graphs of a hyperbola. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci are shown with points that lie on the transverse axis, which is the x-axis. The branches pass through the vertices and open left and right. The y-axis is the conjugate axis. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci lie are shown with points that lie on the transverse axis, which is the y-axis. The branches pass through the vertices and open up and down. The x-axis is the conjugate axis.\" \/><\/span><\/p>\n<p id=\"fs-id1163873631136\">Again our goal is to connect the geometry of a conic with algebra. Placing the hyperbola on a rectangular coordinate system gives us that opportunity. In the figure, we placed the hyperbola so the foci <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-648c27f35a5afa4689612967894ad52a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#99;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#92;&#108;&#101;&#102;&#116;&#40;&#99;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"99\" style=\"vertical-align: -4px;\" \/> are on the <em data-effect=\"italics\">x<\/em>-axis and the center is the origin.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873622834\" data-alt=\"The figure shows the graph of a hyperbola. The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The foci (negative c, 0) and (c, 0) are marked with a point and lie on the x-axis. The vertices are marked with a point and lie on the x-axis. The branches pass through the vertices and open left and right. The distance from (negative c, 0) to a point on the branch (x, y) is marked d sub 1. The distance from (x, y) on the branch to (c, 0) is marked d sub 2.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a hyperbola. The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The foci (negative c, 0) and (c, 0) are marked with a point and lie on the x-axis. The vertices are marked with a point and lie on the x-axis. The branches pass through the vertices and open left and right. The distance from (negative c, 0) to a point on the branch (x, y) is marked d sub 1. The distance from (x, y) on the branch to (c, 0) is marked d sub 2.\" \/><\/span><\/p>\n<p id=\"fs-id1163873764219\">The definition states the difference of the distance from the foci to a point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aee61752ae042431152087f74b766103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#44;&#121;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"39\" style=\"vertical-align: -4px;\" \/> is constant. So <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b7873a2c756cd2e88ae176374256930_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#45;&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"61\" style=\"vertical-align: -4px;\" \/> is a constant that we will call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02baaffe4ddb6a44400eb7ba175e566c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-657ab8816fabb4bcd7e4323ef5385ddb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#45;&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#124;&#61;&#50;&#97;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/> We will use the distance formula to lead us to an algebraic formula for an ellipse.<\/p>\n<p id=\"fs-id1163873857481\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e3e93a637cf0d9cf19d492e239ad91a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#124;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#61;&#50;&#97;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#85;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#115;&#116;&#97;&#110;&#99;&#101;&#32;&#102;&#111;&#114;&#109;&#117;&#108;&#97;&#32;&#116;&#111;&#32;&#102;&#105;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#100;&#125;&#95;&#123;&#49;&#125;&#44;&#123;&#100;&#125;&#95;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#115;&#113;&#114;&#116;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#99;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#124;&#61;&#50;&#97;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#108;&#105;&#109;&#105;&#110;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#114;&#97;&#100;&#105;&#99;&#97;&#108;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#115;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#101;&#108;&#108;&#105;&#112;&#115;&#101;&#44;&#32;&#119;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#99;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#111;&#44;&#32;&#116;&#104;&#101;&#32;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#32;&#111;&#102;&#32;&#97;&#32;&#104;&#121;&#112;&#101;&#114;&#98;&#111;&#108;&#97;&#32;&#99;&#101;&#110;&#116;&#101;&#114;&#101;&#100;&#32;&#97;&#116;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#111;&#114;&#105;&#103;&#105;&#110;&#32;&#105;&#110;&#32;&#115;&#116;&#97;&#110;&#100;&#97;&#114;&#100;&#32;&#102;&#111;&#114;&#109;&#32;&#105;&#115;&#58;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"159\" width=\"864\" style=\"vertical-align: -74px;\" \/><\/p>\n<p id=\"fs-id1163873672757\">To graph the hyperbola, it will be helpful to know about the intercepts. We will find the <em data-effect=\"italics\">x<\/em>-intercepts and <em data-effect=\"italics\">y<\/em>-intercepts using the formula.<\/p>\n<p id=\"fs-id1163873799747\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-014a7e2e8ba8877560434ef49f19908f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#121;&#125;&#125;&#92;&#109;&#97;&#116;&#104;&#98;&#102;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#101;&#116;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#61;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#48;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#120;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#97;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#121;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#32;&#97;&#114;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#114;&#101;&#32;&#97;&#114;&#101;&#32;&#110;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#105;&#110;&#116;&#101;&#114;&#99;&#101;&#112;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"160\" width=\"705\" style=\"vertical-align: -76px;\" \/><\/p>\n<p id=\"fs-id1163873800186\">The <em data-effect=\"italics\">a<\/em>, <em data-effect=\"italics\">b<\/em> values in the equation also help us find the asymptotes of the hyperbola. The asymptotes are intersecting straight lines that the branches of the graph approach but never intersect as the <em data-effect=\"italics\">x<\/em>, <em data-effect=\"italics\">y<\/em> values get larger and larger.<\/p>\n<p id=\"fs-id1163873646422\">To find the asymptotes, we sketch a rectangle whose sides intersect the <em data-effect=\"italics\">x<\/em>-axis at the vertices <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9959f9bc5b535a53482e1f95d8e47160_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and intersect the <em data-effect=\"italics\">y<\/em>-axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6556c12066af78ce791ca65908b97ce5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5bac775607ac99b49b72fd1654a94604_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"44\" style=\"vertical-align: -4px;\" \/> The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola. The rectangle and asymptotes are not part of the hyperbola, but they help us graph the hyperbola.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873809752\" data-alt=\"The figure shows the graph of a hyperbola. The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (negative a, 0) and (a, 0) and are marked with a point and lie on the x-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the x-axis at the vertices (negative a, 0) and (a, 0) and intersect the y-axis at (0, b) and (0, negative b). The asymptotes are given by y is equal to b divided by a times x and y is equal to negative b divided by a times x and are drawn as the diagonals of the central rectangle. The branches of the hyperbola pass through the vertices, open left and right, and approach the asymptotes.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of a hyperbola. The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (negative a, 0) and (a, 0) and are marked with a point and lie on the x-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the x-axis at the vertices (negative a, 0) and (a, 0) and intersect the y-axis at (0, b) and (0, negative b). The asymptotes are given by y is equal to b divided by a times x and y is equal to negative b divided by a times x and are drawn as the diagonals of the central rectangle. The branches of the hyperbola pass through the vertices, open left and right, and approach the asymptotes.\" \/><\/span><\/p>\n<p id=\"fs-id1163869199949\">The asymptotes pass through the origin and we can evaluate their slope using the rectangle we sketched. They have equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0742d1972068d918d239e3c1ba9b08f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"54\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bd029d5a881b6406a2e45a29b4e9a4ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#120;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"72\" style=\"vertical-align: -6px;\" \/><\/p>\n<p id=\"fs-id1163873510103\">There are two equations for hyperbolas, depending whether the transverse axis is vertical or horizontal. We can tell whether the transverse axis is horizontal by looking at the equation. When the equation is in standard form, if the <em data-effect=\"italics\">x<\/em><sup>2<\/sup>-term is positive, the transverse axis is horizontal. When the equation is in standard form, if the <em data-effect=\"italics\">y<\/em><sup>2<\/sup>-term is positive, the transverse axis is vertical.<\/p>\n<p id=\"fs-id1163873744316\">The second equations could be derived similarly to what we have done. We will summarize the results here.<\/p>\n<div data-type=\"note\" id=\"fs-id1163873858397\">\n<div data-type=\"title\">Standard Form of the Equation a Hyperbola with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1163873645567\">The standard form of the equation of a hyperbola with center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-825405fc63416ad0c306970366e996d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/> is<\/p>\n<div data-type=\"equation\" id=\"fs-id1163873668215\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0444fa9494b6c93e915ea2ba94833aee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"265\" style=\"vertical-align: -7px;\" \/><\/div>\n<p><span data-type=\"media\" id=\"fs-id1163870621360\" data-alt=\"The figure shows the graph of two hyperbolas. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (negative a, 0) and (a, 0) and are marked with a point and lie on the x-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the x-axis at the vertices (negative a, 0) and (a, 0) and intersect the y-axis at (0, b) and (0, negative b). The asymptotes are given by y is equal to b divided by a times x and y is equal to negative b divided by a times x and are drawn as the diagonals of the central rectangle. The branches of the hyperbola pass through the vertices, open left and right, and approach the asymptotes. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (0, a) and (0, negative a) and are marked with a point and lie on the y-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the y-axis at the vertices (0, a) and (0, negative a) and intersect the y-axis at (negative b, 0) and (b, 0). The branches of the hyperbola pass through the vertices, open up and down, and approach the asymptotes.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_005_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows the graph of two hyperbolas. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (negative a, 0) and (a, 0) and are marked with a point and lie on the x-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the x-axis at the vertices (negative a, 0) and (a, 0) and intersect the y-axis at (0, b) and (0, negative b). The asymptotes are given by y is equal to b divided by a times x and y is equal to negative b divided by a times x and are drawn as the diagonals of the central rectangle. The branches of the hyperbola pass through the vertices, open left and right, and approach the asymptotes. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices are (0, a) and (0, negative a) and are marked with a point and lie on the y-axis. The points (0, b) and (0, negative) lie on the on the y-axis. There is a central rectangle who sides intersect the y-axis at the vertices (0, a) and (0, negative a) and intersect the y-axis at (negative b, 0) and (b, 0). The branches of the hyperbola pass through the vertices, open up and down, and approach the asymptotes.\" \/><\/span><\/p>\n<p id=\"fs-id1163873663161\">Notice that, unlike the equation of an ellipse, the denominator of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> is not always <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91fa2bda4c48f12d3beccdcacaba4770_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"16\" style=\"vertical-align: 0px;\" \/> and the denominator of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/> is not always <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cce4a9873ea8b9326082bc75fbac9081_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/p>\n<p id=\"fs-id1163873724345\">Notice that when the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/>-term is positive, the transverse axis is on the <em data-effect=\"italics\">x<\/em>-axis. When the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f72e617ab66ab04529ce474aaeeba224_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -4px;\" \/>-term is positive, the transverse axis is on the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<\/div>\n<table id=\"fs-id1163873605544\" summary=\"The table has three columns and eight rows. The first row is a title row and is labeled &#x2018;Standard Forms of the Equation a Hyperbola with Center (0, 0). The second row is a header row with the first column labeled the quantity x squared divided by a squared end quantity minus the quantity y squared divided by b squared is equal to 1 and the second column labeled the quantity y squared divided by a squared end quantity minus the quantity x squared divided by b squared is equal to 1. The rows are labeled &#x2018;Orientation&#x2019;, &#x2018;Vertices&#x2019;, &#x2018;x-intercepts&#x2019;, &#x2018;y-intercepts&#x2019;, &#x2018;Rectangle&#x2019;, and &#x2018;Asymptotes. In row three, the &#x2018;Orientations&#x2019; are &#x2018;transverse axis on the x-axis; opens left and right&#x2019; and &#x2018;transverse axis on the y-axis; opens up and down&#x2019;. In row four, the &#x2018;Vertices&#x2019; are (negative a, 0) and (a, 0) and (0, negative a) and (0, a). In row five, the x-intercepts are (negative a, 0) and (a, 0) and &#x2018;none). In row six, the y-intercepts are &#x2018;none&#x2019; and (0, negative a) and (0, a). In row seven, the rectangle uses (plus or minus a, 0) and (0, plus or minus) and uses (0, plus or minus a) and (plus or minus b, 0). In row eight, the asymptotes are y is equal to b divided by a times x, y is equal to negative b divided by a times x and y is equal to a divided by b times x and y is equal to negative a divided by b times x.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Standard Forms of the Equation a Hyperbola with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"center\"><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33b7e8b6a1f1992f9d09583c4806716b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8348795f8908bdc3454bbcda485bc52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Orientation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Transverse axis on the <em data-effect=\"italics\">x<\/em>-axis.<span data-type=\"newline\"><br \/><\/span>Opens left and right<\/td>\n<td data-valign=\"middle\" data-align=\"center\">Transverse axis on the <em data-effect=\"italics\">y<\/em>-axis.<span data-type=\"newline\"><br \/><\/span>Opens up and down<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Vertices<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9959f9bc5b535a53482e1f95d8e47160_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-039b227d33f66e2c588cbde776e2bb8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-042db233e68dd673e4e2f7cc4d34d1a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9959f9bc5b535a53482e1f95d8e47160_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">none<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">none<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-039b227d33f66e2c588cbde776e2bb8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-042db233e68dd673e4e2f7cc4d34d1a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Rectangle<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-268c31795f618618dee949b58550bcb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59e66dd5659e6a3b7223f6b816100d76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f44190b524443b99af4fc85ac4129681_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-472e60cfb9975dfa696ab545beb90643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#98;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">asymptotes<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6eaaeacbaa96667fffb76ec4f9cf30cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec1100cdf2ae5b1463838e6cfeb2eece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a74e5c569e8dddb3e507dae826fa3fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -6px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3adc5d390e38b7928a61da2c6caa1128_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1163873552620\">We will use these properties to graph hyperbolas.<\/p>\n<div data-type=\"example\" id=\"fs-id1163873626915\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Graph a Hyperbola with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873626917\">\n<div data-type=\"problem\" id=\"fs-id1163873626919\">\n<p id=\"fs-id1163873800346\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b546a599a0ca88b945fadffb45944c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873603471\"><span data-type=\"media\" id=\"fs-id1163873603473\" data-alt=\"Step 1 is to write the equation in standard form. The the quantity x squared divided by 25 end quantity minus the quantity y squared divided by 4 end quantity is equal to 1 is already in standard form.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the equation in standard form. The the quantity x squared divided by 25 end quantity minus the quantity y squared divided by 4 end quantity is equal to 1 is already in standard form.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873814884\" data-alt=\"Step 2 is to determine whether the transverse axis is horizontal or vertical. Since the x squared term is positive, the transverse axis is horizontal.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to determine whether the transverse axis is horizontal or vertical. Since the x squared term is positive, the transverse axis is horizontal.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873606120\" data-alt=\"Step 3 is to find the vertices. Since a squared is equal to 25, then a is equal to plus or minus 5. The vertices lie on the x-axis and are (negative 5, 0) and (5, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to find the vertices. Since a squared is equal to 25, then a is equal to plus or minus 5. The vertices lie on the x-axis and are (negative 5, 0) and (5, 0).\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873899041\" data-alt=\"Step 4 is to sketch the rectangle centered at the origin, intersecting one axis at plus or minus a and the other at plus or minus b. Since a is equal to plus or minus 5, the rectangle will intersect the x-axis at the vertices. Since b is equal to plus or minus 2, the rectangle will intersect the y-axis at (0, negative 2) and (0, 2). The rectangle is shown on a coordinate plane with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to sketch the rectangle centered at the origin, intersecting one axis at plus or minus a and the other at plus or minus b. Since a is equal to plus or minus 5, the rectangle will intersect the x-axis at the vertices. Since b is equal to plus or minus 2, the rectangle will intersect the y-axis at (0, negative 2) and (0, 2). The rectangle is shown on a coordinate plane with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873508574\" data-alt=\"Step 5 is to sketch the asymptotes, the lines through the diagonals of the rectangle. The asymptotes have the equations y is equal to five-halves times x and y is equal to negative five-halves x. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled and the lines that represent the asymptotes.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to sketch the asymptotes, the lines through the diagonals of the rectangle. The asymptotes have the equations y is equal to five-halves times x and y is equal to negative five-halves x. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled and the lines that represent the asymptotes.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873676691\" data-alt=\"Step 6 is to draw the two branches of the hyperbola. Start at each vertex and use the asymptotes as a guide. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled, the lines that represent the asymptotes, y is equal to plus or minus five-halves times x, and the branches that pass through (plus or minus 5, 0) and open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_006f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to draw the two branches of the hyperbola. Start at each vertex and use the asymptotes as a guide. The coordinate plane shows the rectangle with the points (0, 2), (0, negative 2), (negative 5, 0), and (5, 0) labeled, the lines that represent the asymptotes, y is equal to plus or minus five-halves times x, and the branches that pass through (plus or minus 5, 0) and open left and right.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873758424\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873758427\">\n<div data-type=\"problem\" id=\"fs-id1163873770598\">\n<p id=\"fs-id1163873770600\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f41f708cd76e60ae5411a4d63860f4f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"93\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873632906\"><span data-type=\"media\" id=\"fs-id1163873632910\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_302_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873629403\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873629407\">\n<div data-type=\"problem\" id=\"fs-id1163870357696\">\n<p id=\"fs-id1163870357699\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2671926c05a2a6221acdb3f17241c82f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"93\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873644840\"><span data-type=\"media\" id=\"fs-id1163873644843\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus four-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_303_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus four-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873893175\">We summarize the steps for reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1163873893179\" class=\"howto\">\n<div data-type=\"title\">Graph a hyperbola centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/div>\n<ol id=\"fs-id1163873853922\" type=\"1\" class=\"stepwise\">\n<li>Write the equation in standard form.<\/li>\n<li>Determine whether the transverse axis is horizontal or vertical.<\/li>\n<li>Find the vertices.<\/li>\n<li>Sketch the rectangle centered at the origin intersecting one axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b2e6b89ea698237f4882bbc6547aeea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and the other at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec30b859c7891c21c0dd230fc9a5f179_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>Sketch the asymptotes\u2014the lines through the diagonals of the rectangle.<\/li>\n<li>Draw the two branches of the hyperbola.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1163873666021\">Sometimes the equation for a hyperbola needs to be first placed in standard form before we graph it.<\/p>\n<div data-type=\"example\" id=\"fs-id1163873666024\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873557621\">\n<div data-type=\"problem\" id=\"fs-id1163873557623\">\n<p id=\"fs-id1163873557625\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5c1a4963884ae035165f31439c24dfa0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870557545\">\n<table id=\"fs-id1163873862986\" class=\"unnumbered unstyled\" summary=\"4 y squared minus 16 x squared is equal to 64. To write the equation in standard form, divide each term by 64 to make the equation equal to 1. The quantity 4 y divided by 64 end quantity minus 16 x divided by 64 end quantity is equal to the quantity 64 divided by 64. Simplify. The result is the quantity y squared divided by 16 end quantity minus the quantity x squared divided by 4 end quantity is equal to 1. Since the y-squared term is positive, the transverse axis is vertical. Since a squared is equal to 16, then a is equal to plus or minus 4. The vertices lie on the y-axis and are (0, negative a) and (0, a). The vertices are (0, negative 4) and (0, 4). Since b squared is equal to 4, then b is equal to plus or minus 2. Sketch the rectangle intersecting the x-axis at (negative 2, 0) and (2, 0) and the y-axis at the vertices. Sketch the asymptotes through the diagonals of the rectangle. Draw the two branches of the hyperbola. The graph that results is a rectangle that intersects the x-axis at (plus or minus 2, 0) and the y-axis at (0, plus or minus 4), asymptotes that are the diagonals are the rectangle, and branches that pass through the vertices (0, plus or minus 4), and that open up and down.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91fe0a23b497d4ddb2143c6a318b6fad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To write the equation in standard form, divide<span data-type=\"newline\"><br \/><\/span>each term by 64 to make the equation equal to 1.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6c1271ffb8b35d108858a76d006c3503_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#54;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#54;&#52;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#52;&#125;&#123;&#54;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"116\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a187b7e13bbaf70b4fccd2802a438b58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"87\" style=\"vertical-align: -7px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since the <em data-effect=\"italics\">y<\/em><sup>2<\/sup>-term is positive, the transverse axis is vertical.<span data-type=\"newline\"><br \/><\/span>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3f213f16a9dbaf8ce89e93373137bf4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" style=\"vertical-align: -1px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40b83015f2a9c84b73e64577bcdadac3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">The vertices are on the <em data-effect=\"italics\">y<\/em>-axis, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-039b227d33f66e2c588cbde776e2bb8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-daed3d0fc3713237bb6baeb281331921_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6471bd17e56e2362be3b6457d51d8b95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"48\" style=\"vertical-align: -1px;\" \/> then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f854bce5487113b72df9e01bcca8b63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3ed945293025651eb803ad390f45fd18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2c3a69d33f9737210f9c4f1551f4b9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the rectangle intersecting the <em data-effect=\"italics\">x<\/em>-axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4050f6611d38892e64c174797e0a0e29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"59\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-559928bd7c8949c8342dd73437aef05a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> and the <em data-effect=\"italics\">y<\/em>-axis at the vertices.<span data-type=\"newline\"><br \/><\/span>Sketch the asymptotes through the diagonals of the rectangle.<span data-type=\"newline\"><br \/><\/span>Draw the two branches of the hyperbola.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869153025\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163870619489\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163870619492\">\n<div data-type=\"problem\" id=\"fs-id1163873819474\">\n<p id=\"fs-id1163873819476\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4eaea948b1cae73b8b9c5d6976363c3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873604880\"><span data-type=\"media\" id=\"fs-id1163873604883\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_304_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163870504871\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163870504874\">\n<div data-type=\"problem\" id=\"fs-id1163870504876\">\n<p id=\"fs-id1163870504878\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-178ddb234265f79883198cf47a67e8ed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873800252\"><span data-type=\"media\" id=\"fs-id1163873800255\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus three-fifths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_305_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus three-fifths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873912654\">\n<h3 data-type=\"title\">Graph a Hyperbola with Center at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/h3>\n<p id=\"fs-id1163874047291\">Hyperbolas are not always centered at the origin. When a hyperbola is centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> the equations changes a bit as reflected in the table.<\/p>\n<table id=\"fs-id1163873998386\" summary=\"The table has three columns and six rows. The first row is a title row and is labeled &#x2018;Standard Forms of the Equation a Hyperbola with Center (h, k). The second row is a header row with the first column labeled the quantity x minus h squared all divided by a squared end quantity minus the quantity y minus k squared all divided by b squared end quantity is equal to 1 and the second column labeled the quantity y minus k squared all divided by a squared end quantity minus the quantity x minus h squared all divided by b squared end quantity is equal to 1. The rows are labeled &#x2018;Orientation&#x2019;, &#x2018;Center&#x2019;, &#x2018;Vertices&#x2019;, and &#x2018;Rectangle&#x2019;. In row three, the &#x2018;Orientations&#x2019; are &#x2018;transverse axis on the x-axis; opens left and right&#x2019; and &#x2018;transverse axis on the y-axis; opens up and down&#x2019;. In row four, the &#x2018;Centers&#x2019; are both (h, k). In row five, the &#x2018;Vertices&#x2019; are a units to the left and right of the center and a units above and below the center. In row six, the &#x2018;Rectangles&#x2019; are formed by moving a units left or right of the center and b units above or below the center, and by using a units above or below the center and b units left or right of the center.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Standard Forms of the Equation a Hyperbola with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"center\"><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3e21af717b5eca57e15d45e9284308f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"146\" style=\"vertical-align: -7px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ea527f46ff5f9747a6514bed5a4f6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"146\" style=\"vertical-align: -7px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Orientation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Transverse axis is horizontal.<span data-type=\"newline\"><br \/><\/span>Opens left and right<\/td>\n<td data-valign=\"middle\" data-align=\"center\">Transverse axis is vertical.<span data-type=\"newline\"><br \/><\/span>Opens up and down<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Center<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Vertices<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><em data-effect=\"italics\">a<\/em> units to the left and right of the center<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><em data-effect=\"italics\">a<\/em> units above and below the center<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Rectangle<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Use <em data-effect=\"italics\">a<\/em> units left\/right of center<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">b<\/em> units above\/ below the center<\/td>\n<td data-valign=\"middle\" data-align=\"center\">Use <em data-effect=\"italics\">a<\/em> units above\/below the center<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">b<\/em> units left\/right of center<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" id=\"fs-id1163873659370\" class=\"textbox textbox--examples\">\n<div data-type=\"title\">How to Graph a Hyperbola with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873628504\">\n<div data-type=\"problem\" id=\"fs-id1163873628506\">\n<p id=\"fs-id1163869142190\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f14e7d6f7cd913e2a73bc3f74d9b00e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873805530\"><span data-type=\"media\" id=\"fs-id1163873805533\" data-alt=\"Step 1 is to write the equation in standard form. Notice that the equation the quantity x minus 1 squared all divided by 9 end quantity minus the quantity y minus 2 squared all divided by 16 end quantity is equal to 1 is already in standard form.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 1 is to write the equation in standard form. Notice that the equation the quantity x minus 1 squared all divided by 9 end quantity minus the quantity y minus 2 squared all divided by 16 end quantity is equal to 1 is already in standard form.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873665542\" data-alt=\"Step 2 is to deteremine whether the transverse axis is horizonal or vertical. Since the x squared term is positive, the hyperbola opens left and right. The transverse axis is horizontal. The hyperbola opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 2 is to deteremine whether the transverse axis is horizonal or vertical. Since the x squared term is positive, the hyperbola opens left and right. The transverse axis is horizontal. The hyperbola opens left and right.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873769017\" data-alt=\"Step 3 is to find the center and a and b. h is equal to 1 and k is equal 2. a squared is equal to 9 and b squared is equal to 16. You can see tha x minus h is x minus 1, and that y minus k is y minus 2. So, the center is (1, 2) and a is equal to 3 and b is equal to 4.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 3 is to find the center and a and b. h is equal to 1 and k is equal 2. a squared is equal to 9 and b squared is equal to 16. You can see tha x minus h is x minus 1, and that y minus k is y minus 2. So, the center is (1, 2) and a is equal to 3 and b is equal to 4.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873952101\" data-alt=\"Step 4 is to sketch the rectangle centered at (h, k) using a and b. Mark the center (1, 2) on a coordinate plane. Sketch the rectangle that goes through the points 3 units to the left and right of the center and 4 units above and below the center.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 4 is to sketch the rectangle centered at (h, k) using a and b. Mark the center (1, 2) on a coordinate plane. Sketch the rectangle that goes through the points 3 units to the left and right of the center and 4 units above and below the center.\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873863775\" data-alt=\"Step 5 is to sketch the asymptotes on the coordinate plane. They are the lines through the diagonals of the retcangle. Mark the vertices which lie on the rectangle 3 units to the left and right of the center. The vertices are (negative 2, 2) and (4, 2).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 5 is to sketch the asymptotes on the coordinate plane. They are the lines through the diagonals of the retcangle. Mark the vertices which lie on the rectangle 3 units to the left and right of the center. The vertices are (negative 2, 2) and (4, 2).\" \/><\/span><span data-type=\"media\" id=\"fs-id1163873797607\" data-alt=\"Step 6 is to draw the branches of the hyperbola. Start the vertices, (negative 2, 2) and (4, 2) and use the asymptotes as a guide. The branches should open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_008f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"Step 6 is to draw the branches of the hyperbola. Start the vertices, (negative 2, 2) and (4, 2) and use the asymptotes as a guide. The branches should open left and right.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873787440\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873787444\">\n<div data-type=\"problem\" id=\"fs-id1163873787446\">\n<p id=\"fs-id1163873607103\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cbd95a896c2ab9214b3d34e9b9997fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873870786\"><span data-type=\"media\" id=\"fs-id1163873870789\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with an asymptote that passes through (negative 2, negative 2) and (8, 4) and an asymptote that passes through (negative 2, 4) and (8, negative 2), and branches that pass through the vertices (negative 2, 2) and (8, 2) and opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_306_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with an asymptote that passes through (negative 2, negative 2) and (8, 4) and an asymptote that passes through (negative 2, 4) and (8, negative 2), and branches that pass through the vertices (negative 2, 2) and (8, 2) and opens left and right.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873757662\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163870376223\">\n<div data-type=\"problem\" id=\"fs-id1163870376225\">\n<p id=\"fs-id1163870376227\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6fc92bae657535b58a2ef17ee6cd09c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873765727\"><span data-type=\"media\" id=\"fs-id1163873629600\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (2, 2), an asymptote that passes through (0, negative 1) and (4, 5) and an asymptote that passes through (0, 5) and (4, negative 1), and branches that pass through the vertices (0, 2) and (4, 2) and opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_307_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (2, 2), an asymptote that passes through (0, negative 1) and (4, 5) and an asymptote that passes through (0, 5) and (4, negative 1), and branches that pass through the vertices (0, 2) and (4, 2) and opens left and right.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873659703\">We summarize the steps for easy reference.<\/p>\n<div data-type=\"note\" id=\"fs-id1163873659707\" class=\"howto\">\n<div data-type=\"title\">Graph a hyperbola centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/div>\n<ol id=\"fs-id1163873823602\" type=\"1\" class=\"stepwise\">\n<li>Write the equation in standard form.<\/li>\n<li>Determine whether the transverse axis is horizontal or vertical.<\/li>\n<li>Find the center and <em data-effect=\"italics\">a, b<\/em>.<\/li>\n<li>Sketch the rectangle centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> using <em data-effect=\"italics\">a, b<\/em>.<\/li>\n<li>Sketch the asymptotes\u2014the lines through the diagonals of the rectangle. Mark the vertices.<\/li>\n<li>Draw the two branches of the hyperbola.<\/li>\n<\/ol>\n<\/div>\n<p id=\"fs-id1163873623710\">Be careful as you identify the center. The standard equation has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2260901a06586cb13481494f60ade71f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#45;&#104;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7171a455fcdd5dec32692887a52eac22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#45;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"40\" style=\"vertical-align: -4px;\" \/> with the center as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1163873796848\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873796850\">\n<div data-type=\"problem\" id=\"fs-id1163873620826\">\n<p id=\"fs-id1163873620828\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6d4d3f15a7c437d0152c4e322fbb9447_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870292411\">\n<table id=\"fs-id1163873864599\" class=\"unnumbered unstyled\" summary=\"The quantity y plus 2 squared all divided by 9 end quantity minus the quantity x plus 1 all divided by 4 end quantity is equal to 1. Since the y squared term is positive, the hyperbola opens up and down. The expression y minus k is given by y minus negative 2 and the expression x minus h is given by x minus negative 1. Find the center, (h, k). Find a and b. The center is (negative 1, negative 2) and a is equal to 3 and b is equal to 2. Sketch the rectangle that goes through the points 3 units above and below the center and 2 units to the left and right of the center. Sketch the asymptotes, the lines through the diagonals of the rectangle. One asymptote passes through (negative 3, negative 5) and (1, 1) and the other passes through (negative 3, 1) and (1, negative 5) Mark the vertices. Graph the branches, which pass through the vertices (negative 3, negative 2) and (1, negative 2) and open up and down.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870345938\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/>term is positive, the hyperbola<span data-type=\"newline\"><br \/><\/span>opens up and down.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873703032\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ce38ace23dd1b8f15095a1aff4a73f36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <em data-effect=\"italics\">a, b<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9418f76b7fb8efbd61d4b14b3df06bad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41863e650091732c7785e84b2c83252a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#98;&#61;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"39\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the rectangle that goes through the<span data-type=\"newline\"><br \/><\/span>points 3 units above and below the center and<span data-type=\"newline\"><br \/><\/span>2 units to the left\/right of the center.<span data-type=\"newline\"><br \/><\/span>Sketch the asymptotes\u2014the lines through the<span data-type=\"newline\"><br \/><\/span>diagonals of the rectangle.<span data-type=\"newline\"><br \/><\/span>Mark the vertices.<span data-type=\"newline\"><br \/><\/span>Graph the branches.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163869585742\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873732322\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163870254062\">\n<div data-type=\"problem\" id=\"fs-id1163870254064\">\n<p id=\"fs-id1163870254067\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2e4f027593365efce242b80d2fbc0ce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"149\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873861041\"><span data-type=\"media\" id=\"fs-id1163873861044\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 3), an asymptote that passes through (negative 5, negative 7) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, 7), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 7) and opens up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_308_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 3), an asymptote that passes through (negative 5, negative 7) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, 7), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 7) and opens up and down.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873558405\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873789429\">\n<div data-type=\"problem\" id=\"fs-id1163873789431\">\n<p id=\"fs-id1163873789433\">Graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4b5fe8f60f8260de364472fc62a015f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"149\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873884826\"><span data-type=\"media\" id=\"fs-id1163873605470\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 2), an asymptote that passes through (negative 5, negative 5) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, negative 5), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 5) and opens up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_309_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with a center at (negative 2, negative 2), an asymptote that passes through (negative 5, negative 5) and (1, 1) and an asymptote that passes through (negative 5, 1) and (1, negative 5), and branches that pass through the vertices (negative 2, 1) and (negative 2, negative 5) and opens up and down.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1163867181069\">Again, sometimes we have to put the equation in standard form as our first step.<\/p>\n<div data-type=\"example\" id=\"fs-id1163873743007\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873743009\">\n<div data-type=\"problem\" id=\"fs-id1163873743011\">\n<p id=\"fs-id1163873743013\">Write the equation in standard form and graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b82ea423112099fbf39eeb00814db25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#120;&#45;&#51;&#54;&#121;&#45;&#51;&#54;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"249\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870548674\">\n<table id=\"fs-id1163870548677\" class=\"unnumbered unstyled can-break\" summary=\"4 x squared minus 9 y squared minus 24 x minus 36 y minus 36 is equal to 0. To get to standard form, complete the squares. 4 times the quantity x squared minus 6 x blank end quantity minus 9 times the quantity y squared plus 4 y blank) end quantity is equal to 36. 4 times the quantity x squared minus 6 x plus 9 end quantity minus 9 times the quantity y squared plus 4 y plus 4 end quantity is equal to 36 plus 36 minus 36. 4 times the quantity x minus 3 squared minus 9 times the quantity y plus 2 squared is equal to 36. Divide each term by 36 to get the constant to be 1. 4 times the quantity x minus 3 squared all divided by 36 minus 9 times the quantity y plus 2 squared all divided by 36 is equal to 36 divided by 36. The result is the quantity x minus 3 squared all divided by 36 minus the quantity y plus 2 squared all divided by 4 is equal to 1. Since the x squared term is positive, the hyperbola opens left and right. Find the center, (h, k). The center is (3, negative 2). Find a and b. a is equal to 3 and b is equal to 4. Sketch the rectangle that goes through the points 3 units to the left and right of the center and 2 units above and below the center. Sketch the asymptotes, the lines through the diagonals of the rectangle. One asymptote passes through (0, 0) and (6, negative 4) and the other passes through (0, negative 4) and (6, 0). Mark the vertices at (0, negative 2) and (6, negative 2). Graph the branches, making sure that they pass through the vertices.\" data-label=\"\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873791834\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">To get to standard form, complete the squares.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873648496\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163870463090\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873764283\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Divide each term by 36 to get the constant to be 1.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163867240277\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\"><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873792475\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Since the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/>term is positive, the hyperbola<span data-type=\"newline\"><br \/><\/span>opens left and right.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find the center, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">Center: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dbaec542907415eac32615dfae0ae911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#44;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"52\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Find <em data-effect=\"italics\">a, b<\/em>.<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d9db73cc7d54e793e9cccce32a54d30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#97;&#61;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#98;&#61;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"42\" style=\"vertical-align: -12px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"left\">Sketch the rectangle that goes through the<span data-type=\"newline\"><br \/><\/span>points 3 units to the left\/right of the center<span data-type=\"newline\"><br \/><\/span>and 2 units above and below the center.<span data-type=\"newline\"><br \/><\/span>Sketch the asymptotes\u2014the lines through the<span data-type=\"newline\"><br \/><\/span>diagonals of the rectangle.<span data-type=\"newline\"><br \/><\/span>Mark the vertices.<span data-type=\"newline\"><br \/><\/span>Graph the branches.<\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1163873644340\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_010g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163870291828\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873626188\">\n<div data-type=\"problem\" id=\"fs-id1163873626190\">\n<p id=\"fs-id1163873626192\"><span class=\"token\">\u24d0<\/span> Write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba41cc64ae4c3afbe87bd26238b12e22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#54;&#52;&#121;&#45;&#49;&#57;&#57;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"266\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870276462\">\n<p id=\"fs-id1163873799558\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e2595a7f1ab2088ece3fbad1ae65d8de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873784873\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (negative 1, 2), an asymptote that passes through (negative 5, 5) and (3, negative 1) and an asymptote that passes through (3, 5) and (negative 5, negative 1), and branches that pass through the vertices (negative 5, 2) and (3, 2) and opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_310_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (negative 1, 2), an asymptote that passes through (negative 5, 5) and (3, negative 1) and an asymptote that passes through (3, 5) and (negative 5, negative 1), and branches that pass through the vertices (negative 5, 2) and (3, 2) and opens left and right.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163869435918\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163869435921\">\n<div data-type=\"problem\" id=\"fs-id1163869435923\">\n<p id=\"fs-id1163869435925\"><span class=\"token\">\u24d0<\/span> Write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71764137dcd23e831c9380be7f5a83b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#54;&#120;&#45;&#53;&#48;&#121;&#45;&#50;&#56;&#49;&#61;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"274\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873872203\">\n<p id=\"fs-id1163873872205\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-55c17c5078fa211dfdffb257413a9720_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873803299\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (negative 3, negative 1), an asymptote that passes through (negative 8, negative 5) and (2, 3) and an asymptote that passes through (negative 8, 3) and (2, negative 5), and branches that pass through the vertices (negative 8, negative 1) and (2, negative 1) and opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_311_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with the center (negative 3, negative 1), an asymptote that passes through (negative 8, negative 5) and (2, 3) and an asymptote that passes through (negative 8, 3) and (2, negative 5), and branches that pass through the vertices (negative 8, negative 1) and (2, negative 1) and opens left and right.\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1163873662594\">\n<h3 data-type=\"title\">Identify Conic Sections by their Equations<\/h3>\n<p id=\"fs-id1163873662599\">Now that we have completed our study of the conic sections, we will take a look at the different equations and recognize some ways to identify a conic by its equation. When we are given an equation to graph, it is helpful to identify the conic so we know what next steps to take.<\/p>\n<p id=\"fs-id1163870414147\">To identify a conic from its equation, it is easier if we put the variable terms on one side of the equation and the constants on the other.<\/p>\n<table id=\"fs-id1163870414151\" summary=\"This table has three columns and five rows. The first row is a header row and it labels each column, &#x201c;Conic,&#x201d; &#x201c;Characteristics of x squared and y squared terms,&#x201d; and &#x201c;Example.&#x201d; The first column is a header column and it labels each row &#x201c;Parabola,&#x201d; &#x201c;Circle,&#x201d; &#x201c;Ellipse,&#x201d;, and &#x201c;Hyperbola.&#x201d; In row two, the Parabola is described as having either x squared or y squared and only one variable squared and the example is x is equal to 3 y squared minus 2 y plus 1. In row three, the Circle is described as having x squared and y squared terms with the same coefficients and the example is x squared plus y squared is equal to 49. In row four, the Ellipse is described as having x squared and y squared terms that have the same sign and different coefficients and the example is 4 x squared plus 25 y squared is equal to 100. In row five, the Hyperbola is described as having x squared and y squared terms that have different signs and different coefficients and the example is 25 y squared minus 4 x squared is equal to 100.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"left\">Conic<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Characteristics of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> terms<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Parabola<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Either <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> OR <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4480113eb9aa667d66baae5c5bfd869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"21\" style=\"vertical-align: -4px;\" \/> Only one variable is squared.<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7759298631d48d9ad0f6b5f7b884e87c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Circle<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> terms have the same coefficients<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4846d3ba91620499c56c560ff0529ae1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Ellipse<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> terms have the <strong data-effect=\"bold\">same<\/strong> sign, different coefficients<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6986ae4e74070667f81f5c59ab374612_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Hyperbola<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> terms have <strong data-effect=\"bold\">different<\/strong> signs, different coefficients<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcd305ea016f44e393d6a78163bad103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" id=\"fs-id1163873621465\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1163873621467\">\n<div data-type=\"problem\" id=\"fs-id1163873621469\">\n<p id=\"fs-id1163873621471\">Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.<\/p>\n<p id=\"fs-id1163873655251\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ed9d7f2b85b31a8511421f432d69767d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#54;&#121;&#43;&#49;&#54;&#48;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94561e7b5ad13f331b4e011f7b3eded2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#54;&#52;&#121;&#45;&#49;&#57;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"262\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-931ce0c76f78044a842dcaef4cc3486f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#56;&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"170\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7c37bf860211fa12559f95c82e2698a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870346275\">\n<p id=\"fs-id1163873599316\"><span class=\"token\">\u24d0<\/span><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-161f41942d58635a6720959bb715379e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#54;&#121;&#43;&#49;&#54;&#48;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#116;&#101;&#114;&#109;&#115;&#32;&#104;&#97;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#115;&#105;&#103;&#110;&#32;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#99;&#111;&#101;&#102;&#102;&#105;&#99;&#105;&#101;&#110;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#108;&#108;&#105;&#112;&#115;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"714\" style=\"vertical-align: -23px;\" \/><span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b2c83926ef05b670287e5f9d43bec36d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#120;&#43;&#54;&#52;&#121;&#45;&#49;&#57;&#57;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#116;&#101;&#114;&#109;&#115;&#32;&#104;&#97;&#118;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#115;&#105;&#103;&#110;&#115;&#32;&#97;&#110;&#100;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#116;&#32;&#99;&#111;&#101;&#102;&#102;&#105;&#99;&#105;&#101;&#110;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#52;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#121;&#112;&#101;&#114;&#98;&#111;&#108;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"60\" width=\"739\" style=\"vertical-align: -23px;\" \/><span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4dc1c5fb9d4933c1dcb6f027fc63d188_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#54;&#120;&#45;&#56;&#121;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#110;&#100;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#116;&#101;&#114;&#109;&#115;&#32;&#104;&#97;&#118;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#32;&#99;&#111;&#101;&#102;&#102;&#105;&#99;&#105;&#101;&#110;&#116;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#105;&#114;&#99;&#108;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"686\" style=\"vertical-align: -15px;\" \/><span data-type=\"newline\"><br \/><\/span><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-afca9a413ea065d73728d8cd2bf3c322_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#121;&#61;&#45;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#110;&#108;&#121;&#32;&#111;&#110;&#101;&#32;&#118;&#97;&#114;&#105;&#97;&#98;&#108;&#101;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#120;&#44;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#105;&#115;&#32;&#115;&#113;&#117;&#97;&#114;&#101;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#114;&#97;&#98;&#111;&#108;&#97;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"41\" width=\"655\" style=\"vertical-align: -15px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163866895479\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873744525\">\n<div data-type=\"problem\" id=\"fs-id1163873744527\">\n<p id=\"fs-id1163873744529\">Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.<\/p>\n<p id=\"fs-id1163873744532\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f890a19822f95a926eb0286c12cb7cab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#56;&#120;&#45;&#54;&#121;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"170\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6986ae4e74070667f81f5c59ab374612_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ad1ac3f62be96cfdc25626aeb4c3625d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"130\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e95371812658f9aecdbc490eb48874f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873650679\">\n<p id=\"fs-id1163873650681\"><span class=\"token\">\u24d0<\/span> circle <span class=\"token\">\u24d1<\/span> ellipse <span class=\"token\">\u24d2<\/span> parabola <span class=\"token\">\u24d3<\/span> hyperbola<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163870407092\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1163873861534\">\n<div data-type=\"problem\" id=\"fs-id1163873861536\">\n<p id=\"fs-id1163873861538\">Identify the graph of each equation as a circle, parabola, ellipse, or hyperbola.<\/p>\n<p id=\"fs-id1163873861542\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d2704801c5f75b76e0a63301dcdc330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ac45acf7f38a404c45ce6901a30c05d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#50;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#52;&#120;&#43;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"131\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19ef381ddc9972ad7e8dea3d2c35fffe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#120;&#43;&#54;&#121;&#43;&#57;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"200\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80459c7e50f217d75ad42f1f7e42496e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873667027\">\n<p id=\"fs-id1163873667029\"><span class=\"token\">\u24d0<\/span> ellipse <span class=\"token\">\u24d1<\/span> parabola <span class=\"token\">\u24d2<\/span> circle <span class=\"token\">\u24d3<\/span> hyperbola<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1163873606913\" class=\"media-2\">\n<p id=\"fs-id1163873606917\">Access these online resources for additional instructions and practice with hyperbolas.<\/p>\n<ul id=\"fs-id1163873606921\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37ghyperborig\">Graph a Hyperbola with Center at the Origin<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37ghyperbnorig\">Graph a Hyperbola with Center not at the Origin<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37ghyperbgen\">Graph a Hyperbola in General Form<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37conicsgen\">Identifying Conic Sections in General Form<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163873870675\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1163873793083\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Hyperbola:<\/strong> A <strong data-effect=\"bold\">hyperbola<\/strong> is all points in a plane where the difference of their distances from two fixed points is constant.<span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1163869200311\" data-alt=\"The figure shows a double napped right circular cone sliced by a plane that is parallel to the vertical axis of the cone forming a hyperbola. The figure is labeled &#x2018;hyperbola&#x2019;.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_011_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows a double napped right circular cone sliced by a plane that is parallel to the vertical axis of the cone forming a hyperbola. The figure is labeled &#x2018;hyperbola&#x2019;.\" \/><\/span><span data-type=\"newline\"><br \/><\/span> Each of the fixed points is called a <strong data-effect=\"bold\">focus<\/strong> of the hyperbola.<span data-type=\"newline\"><br \/><\/span> The line through the foci, is called the <strong data-effect=\"bold\">transverse axis<\/strong>.<span data-type=\"newline\"><br \/><\/span> The two points where the transverse axis intersects the hyperbola are each a <strong data-effect=\"bold\">vertex<\/strong> of the hyperbola.<span data-type=\"newline\"><br \/><\/span> The midpoint of the segment joining the foci is called the <strong data-effect=\"bold\">center<\/strong> of the hyperbola.<span data-type=\"newline\"><br \/><\/span> The line perpendicular to the transverse axis that passes through the center is called the <strong data-effect=\"bold\">conjugate axis<\/strong>.<span data-type=\"newline\"><br \/><\/span> Each piece of the graph is called a <strong data-effect=\"bold\">branch<\/strong> of the hyperbola.<span data-type=\"newline\"><br \/><\/span> <span data-type=\"media\" id=\"fs-id1163873652586\" data-alt=\"The figure shows two graphs of a hyperbola. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci are shown with points that lie on the transverse axis, which is the x-axis. The branches pass through the vertices and open left and right. The y-axis is the conjugate axis. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci lie are shown with points that lie on the transverse axis, which is the y-axis. The branches pass through the vertices and open up and down. The x-axis is the conjugate axis.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_012_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The figure shows two graphs of a hyperbola. The first graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci are shown with points that lie on the transverse axis, which is the x-axis. The branches pass through the vertices and open left and right. The y-axis is the conjugate axis. The second graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals. The center of the hyperbola is the origin. The vertices and foci lie are shown with points that lie on the transverse axis, which is the y-axis. The branches pass through the vertices and open up and down. The x-axis is the conjugate axis.\" \/><\/span><span data-type=\"newline\"><br \/><\/span><br \/>\n<table id=\"fs-id1163873677570\" summary=\"The table has three columns and eight rows. The first row is a title row and is labeled &#x2018;Standard Forms of the Equation a Hyperbola with Center (0, 0). The second row is a header row with the first column labeled the quantity x squared divided by a squared end quantity minus the quantity y squared divided by b squared is equal to 1 and the second column labeled the quantity y squared divided by a squared end quantity minus the quantity x squared divided by b squared is equal to 1. The rows are labeled &#x2018;Orientation&#x2019;, &#x2018;Vertices&#x2019;, &#x2018;x-intercepts&#x2019;, &#x2018;y-intercepts&#x2019;, &#x2018;Rectangle&#x2019;, and &#x2018;Asymptotes. In row three, the &#x2018;Orientations&#x2019; are &#x2018;transverse axis on the x-axis; opens left and right&#x2019; and &#x2018;transverse axis on the y-axis; opens up and down&#x2019;. In row four, the &#x2018;Vertices&#x2019; are (negative a, 0) and (a, 0) and (0, negative a) and (0, a). In row five, the x-intercepts are (negative a, 0) and (a, 0) and &#x2018;none). In row six, the y-intercepts are &#x2018;none&#x2019; and (0, negative a) and (0, a). In row seven, the rectangle uses (plus or minus a, 0) and (0, plus or minus) and uses (0, plus or minus a) and (plus or minus b, 0). In row eight, the asymptotes are y is equal to b divided by a times x, y is equal to negative b divided by a times x and y is equal to a divided by b times x and y is equal to negative a divided by b times x.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Standard Forms of the Equation a Hyperbola with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"center\"><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-33b7e8b6a1f1992f9d09583c4806716b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8348795f8908bdc3454bbcda485bc52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Orientation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Transverse axis on the <em data-effect=\"italics\">x<\/em>-axis.<span data-type=\"newline\"><br \/><\/span>Opens left and right<\/td>\n<td data-valign=\"middle\" data-align=\"center\">Transverse axis on the <em data-effect=\"italics\">y<\/em>-axis.<span data-type=\"newline\"><br \/><\/span>Opens up and down<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Vertices<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9959f9bc5b535a53482e1f95d8e47160_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-039b227d33f66e2c588cbde776e2bb8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-042db233e68dd673e4e2f7cc4d34d1a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">x<\/em>-intercepts<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9959f9bc5b535a53482e1f95d8e47160_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"46\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53d4347201ab8f9c6f195eeec4b01f0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">none<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><em data-effect=\"italics\">y<\/em>-intercepts<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">none<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7069099fec7f8f1d2b63efd79849e18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-042db233e68dd673e4e2f7cc4d34d1a0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Rectangle<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-268c31795f618618dee949b58550bcb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#97;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59e66dd5659e6a3b7223f6b816100d76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#98;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f44190b524443b99af4fc85ac4129681_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#97;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-472e60cfb9975dfa696ab545beb90643_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#98;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"36\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">asymptotes<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6eaaeacbaa96667fffb76ec4f9cf30cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"58\" style=\"vertical-align: -6px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec1100cdf2ae5b1463838e6cfeb2eece_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#98;&#125;&#123;&#97;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a74e5c569e8dddb3e507dae826fa3fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"58\" style=\"vertical-align: -6px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3adc5d390e38b7928a61da2c6caa1128_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#97;&#125;&#123;&#98;&#125;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"68\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><strong data-effect=\"bold\">How to graph a hyperbola centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9b349d335879ab45d8b79d5850b0860_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"45\" style=\"vertical-align: -4px;\" \/><\/strong>\n<ol id=\"fs-id1163873898656\" type=\"1\" class=\"stepwise\">\n<li>Write the equation in standard form.<\/li>\n<li>Determine whether the transverse axis is horizontal or vertical.<\/li>\n<li>Find the vertices.<\/li>\n<li>Sketch the rectangle centered at the origin intersecting one axis at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b2e6b89ea698237f4882bbc6547aeea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"9\" style=\"vertical-align: 0px;\" \/> and the other at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ec30b859c7891c21c0dd230fc9a5f179_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&plusmn;&#125;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"12\" style=\"vertical-align: 0px;\" \/><\/li>\n<li>Sketch the asymptotes\u2014the lines through the diagonals of the rectangle.<\/li>\n<li>Draw the two branches of the hyperbola.<\/li>\n<\/ol>\n<p><span data-type=\"newline\"><br \/><\/span><\/p>\n<table id=\"fs-id1163873632917\" summary=\"The table has three columns and six rows. The first row is a title row and is labeled &#x2018;Standard Forms of the Equation a Hyperbola with Center (h, k). The second row is a header row with the first column labeled the quantity x minus h squared all divided by a squared end quantity minus the quantity y minus k squared all divided by b squared end quantity is equal to 1 and the second column labeled the quantity y minus k squared all divided by a squared end quantity minus the quantity x minus h squared all divided by b squared end quantity is equal to 1. The rows are labeled &#x2018;Orientation&#x2019;, &#x2018;Center&#x2019;, &#x2018;Vertices&#x2019;, and &#x2018;Rectangle&#x2019;. In row three, the &#x2018;Orientations&#x2019; are &#x2018;transverse axis on the x-axis; opens left and right&#x2019; and &#x2018;transverse axis on the y-axis; opens up and down&#x2019;. In row four, the &#x2018;Centers&#x2019; are both (h, k). In row five, the &#x2018;Vertices&#x2019; are a units to the left and right of the center and a units above and below the center. In row six, the &#x2018;Rectangles&#x2019; are formed by moving a units left or right of the center and b units above or below the center, and by using a units above or below the center and b units left or right of the center.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th colspan=\"3\" data-valign=\"middle\" data-align=\"center\">Standard Forms of the Equation a Hyperbola with Center <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/th>\n<\/tr>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"center\"><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3e21af717b5eca57e15d45e9284308f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"146\" style=\"vertical-align: -7px;\" \/><\/th>\n<th data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1ea527f46ff5f9747a6514bed5a4f6fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#97;&#125;&#94;&#123;&#50;&#125;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#104;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#123;&#98;&#125;&#94;&#123;&#50;&#125;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"146\" style=\"vertical-align: -7px;\" \/><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Orientation<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Transverse axis is horizontal.<span data-type=\"newline\"><br \/><\/span>Opens left and right<\/td>\n<td data-valign=\"middle\" data-align=\"center\">Transverse axis is vertical.<span data-type=\"newline\"><br \/><\/span>Opens up and down<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Center<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Vertices<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><em data-effect=\"italics\">a<\/em> units to the left and right of the center<\/td>\n<td data-valign=\"middle\" data-align=\"center\"><em data-effect=\"italics\">a<\/em> units above and below the center<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">Rectangle<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">Use <em data-effect=\"italics\">a<\/em> units left\/right of center<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">b<\/em> units above\/below the center<\/td>\n<td data-valign=\"middle\" data-align=\"center\">Use <em data-effect=\"italics\">a<\/em> units above\/below the center<span data-type=\"newline\"><br \/><\/span><em data-effect=\"italics\">b<\/em> units left\/right of center<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><strong data-effect=\"bold\">How to graph a hyperbola centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfb5576bd61adfab422c523cc9ec93e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"48\" style=\"vertical-align: -4px;\" \/><\/strong>\n<ol id=\"fs-id1163873664094\" type=\"1\" class=\"stepwise\">\n<li>Write the equation in standard form.<\/li>\n<li>Determine whether the transverse axis is horizontal or vertical.<\/li>\n<li>Find the center and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e8e1a6321dcdb6dccc212d72c5f0947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"29\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Sketch the rectangle centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/> using <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e8e1a6321dcdb6dccc212d72c5f0947_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#97;&#44;&#98;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"29\" style=\"vertical-align: -4px;\" \/><\/li>\n<li>Sketch the asymptotes\u2014the lines through the diagonals of the rectangle. Mark the vertices.<\/li>\n<li>Draw the two branches of the hyperbola.<\/li>\n<\/ol>\n<p><span data-type=\"newline\"><br \/><\/span><\/p>\n<table id=\"fs-id1163873558902\" summary=\"This table has three columns and five rows. The first row is a header row and it labels each column, &#x201c;Conic,&#x201d; &#x201c;Characteristics of x squared and y squared terms,&#x201d; and &#x201c;Example.&#x201d; The first column is a header column and it labels each row &#x201c;Parabola,&#x201d; &#x201c;Circle,&#x201d; &#x201c;Ellipse,&#x201d;, and &#x201c;Hyperbola.&#x201d; In row two, the Parabola is described as having either x squared or y squared and only one variable squared and the example is x is equal to 3 y squared minus 2 y plus 1. In row three, the Circle is described as having x squared and y squared terms with the same coefficients and the example is x squared plus y squared is equal to 49. In row four, the Ellipse is described as having x squared and y squared terms that have the same sign and different coefficients and the example is 4 x squared plus 25 y squared is equal to 100. In row five, the Hyperbola is described as having x squared and y squared terms that have different signs and different coefficients and the example is 25 y squared minus 4 x squared is equal to 100.\" class=\"unnumbered\" data-label=\"\">\n<thead>\n<tr valign=\"top\">\n<th data-valign=\"middle\" data-align=\"left\">Conic<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Characteristics of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> terms<\/th>\n<th data-valign=\"middle\" data-align=\"left\">Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Parabola<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\">Either <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b40448f90dbf1bf9cce1035e2f3b1120_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: 0px;\" \/> OR <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b4480113eb9aa667d66baae5c5bfd869_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"21\" style=\"vertical-align: -4px;\" \/> Only one variable is squared.<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7759298631d48d9ad0f6b5f7b884e87c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#51;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Circle<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> terms have the same coefficients<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4846d3ba91620499c56c560ff0529ae1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Ellipse<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> terms have the <strong data-effect=\"bold\">same<\/strong> sign, different coefficients<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6986ae4e74070667f81f5c59ab374612_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"left\"><strong data-effect=\"bold\">Hyperbola<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a3c7ca4039574ac3bde410181a9ffd79_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"23\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2073e2d642509d669c129c9c50c7f49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"22\" style=\"vertical-align: -4px;\" \/> terms have <strong data-effect=\"bold\">different<\/strong> signs, different coefficients<\/td>\n<td data-valign=\"middle\" data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fcd305ea016f44e393d6a78163bad103_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1163873814241\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1163873782017\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1163873782024\"><strong data-effect=\"bold\">Graph a Hyperbola with Center at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c53627fd7039dcb62c54d86fe468e6e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#44;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"38\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1163874053963\">In the following exercises, graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163874053966\">\n<div data-type=\"problem\" id=\"fs-id1163873513529\">\n<p id=\"fs-id1163873513531\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a5b93a9c6611bf81f82dfedd7bde84fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870368930\"><span data-type=\"media\" id=\"fs-id1163870368933\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus two-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_312_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions, but at unlabeled intervals, with asymptotes y is equal to plus or minus two-thirds times x, and branches that pass through the vertices (plus or minus 3, 0) and open left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163874047591\">\n<div data-type=\"problem\" id=\"fs-id1163874047593\">\n<p id=\"fs-id1163874047595\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-df7b23768af1f23ca1a6e28a1350a219_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873657989\">\n<div data-type=\"problem\" id=\"fs-id1163873657991\">\n<p id=\"fs-id1163873657993\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8609711fdf17775fa2ab38aee8407583_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163874047544\"><span data-type=\"media\" id=\"fs-id1163874047547\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-fourths times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_314_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-fourths times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873850658\">\n<div data-type=\"problem\" id=\"fs-id1163870549374\">\n<p id=\"fs-id1163870549376\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c8088755a549785a01d9cc976eb614ec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163874019191\">\n<div data-type=\"problem\" id=\"fs-id1163874019193\">\n<p id=\"fs-id1163873651595\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7efe07fe96430fa64ecf88fe7c2c6c0a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873725271\"><span data-type=\"media\" id=\"fs-id1163873678231\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_316_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus five-halves times x, and branches that pass through the vertices (0, plus or minus 5) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873627622\">\n<div data-type=\"problem\" id=\"fs-id1163873627625\">\n<p id=\"fs-id1163873627627\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56112d5490ad6dfcf389c9f0f75f4d37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163869138006\">\n<div data-type=\"problem\" id=\"fs-id1163869138008\">\n<p id=\"fs-id1163869138010\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e95371812658f9aecdbc490eb48874f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163869142597\"><span data-type=\"media\" id=\"fs-id1163867248314\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-fourths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_318_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-fourths times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870384297\">\n<div data-type=\"problem\" id=\"fs-id1163870384299\">\n<p id=\"fs-id1163870384301\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83a916ec7d5a3904a80db2d7b45abf87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873787573\">\n<div data-type=\"problem\" id=\"fs-id1163873787575\">\n<p id=\"fs-id1163870218685\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b07cfad0f7527882bfaebff89aef792_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873506044\"><span data-type=\"media\" id=\"fs-id1163873506047\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-halves times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_320_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus three-halves times x, and branches that pass through the vertices (0, plus or minus 3) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163869200126\">\n<div data-type=\"problem\" id=\"fs-id1163869200128\">\n<p id=\"fs-id1163869200130\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ebf59d850f2e4ad91a5f211fc596fcd2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#53;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#61;&#52;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873793246\">\n<div data-type=\"problem\" id=\"fs-id1163873793248\">\n<p id=\"fs-id1163873793251\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-80459c7e50f217d75ad42f1f7e42496e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#54;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873802105\"><span data-type=\"media\" id=\"fs-id1163873702446\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_322_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with asymptotes y is equal to plus or minus one-half times x, and branches that pass through the vertices (plus or minus 4, 0) and open left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873814481\">\n<div data-type=\"problem\" id=\"fs-id1163873814483\">\n<p id=\"fs-id1163873814485\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8d274352b0bf39b45280271cb392adce_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#51;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163869406009\"><strong data-effect=\"bold\">Graph a Hyperbola with Center at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-772b77341e52468c6e31b5bff8f72528_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#104;&#44;&#107;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -4px;\" \/><\/strong><\/p>\n<p id=\"fs-id1163873898758\">In the following exercises, graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163873898761\">\n<div data-type=\"problem\" id=\"fs-id1163873627636\">\n<p id=\"fs-id1163873627638\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d74e7ef2dea95d3c7a50e4c68a28365e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163869164198\"><span data-type=\"media\" id=\"fs-id1163869164201\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_324_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873655036\">\n<div data-type=\"problem\" id=\"fs-id1163873655038\">\n<p id=\"fs-id1163873633345\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8a8da8781311a271a26a3a789f95a64e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873607551\">\n<div data-type=\"problem\" id=\"fs-id1163873607553\">\n<p id=\"fs-id1163873607555\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-279bd955414d8b2778cf03daec42b864_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873599765\"><span data-type=\"media\" id=\"fs-id1163873599768\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_326_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 3) an asymptote that passes through (negative 3, 1) and (5, 5) and an asymptote that passes through (5, 1) and (negative 3, 5), and branches that pass through the vertices (negative 3, 3) and (5, 3) and opens left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163874032597\">\n<div data-type=\"problem\" id=\"fs-id1163874032599\">\n<p id=\"fs-id1163874032602\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-88e805a1386b23d7409b7c9e89e7aff4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870462018\">\n<div data-type=\"problem\" id=\"fs-id1163870462020\">\n<p id=\"fs-id1163870462022\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c204234460820c5dd42a0baf79e7c1cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163867240292\"><span data-type=\"media\" id=\"fs-id1163867240295\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, negative 4) an asymptote that passes through (negative 7, 1) and (5, negative 9) and an asymptote that passes through (5, 1) and (negative 7, negative 9), and branches that pass through the vertices (1, 1) and (1, negative 9) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_328_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, negative 4) an asymptote that passes through (negative 7, 1) and (5, negative 9) and an asymptote that passes through (5, 1) and (negative 7, negative 9), and branches that pass through the vertices (1, 1) and (1, negative 9) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873768786\">\n<div data-type=\"problem\" id=\"fs-id1163873768789\">\n<p id=\"fs-id1163873768791\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c2e12be96b9fbb70f565c76ad412b532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873639294\">\n<div data-type=\"problem\" id=\"fs-id1163873639296\">\n<p id=\"fs-id1163869408283\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e92843e54606d2945ccb33eca4843587_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873664555\"><span data-type=\"media\" id=\"fs-id1163873731499\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 1, 4) an asymptote that passes through (4, 8) and (negative 6, 0) and an asymptote that passes through (negative 6, 8) and (4, 0), and branches that pass through the vertices (negative 1, 0) and (negative 1, 8) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_330_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 1, 4) an asymptote that passes through (4, 8) and (negative 6, 0) and an asymptote that passes through (negative 6, 8) and (4, 0), and branches that pass through the vertices (negative 1, 0) and (negative 1, 8) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873753634\">\n<div data-type=\"problem\" id=\"fs-id1163873753636\">\n<p id=\"fs-id1163873753638\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8ae74422e824d513a494a69861563264_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870489128\">\n<div data-type=\"problem\" id=\"fs-id1163870489131\">\n<p id=\"fs-id1163870489133\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04ceb41da8b03c2d0ef6835bec631f13_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873664272\"><span data-type=\"media\" id=\"fs-id1163873664275\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (3, negative 2) an asymptote that passes through (8, 1) and (negative 2, negative 5) and an asymptote that passes through (negative 2, negative 1) and (8, negative 5), and branches that pass through the vertices (negative 2, negative 2) and (8, negative 2) and opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_332_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (3, negative 2) an asymptote that passes through (8, 1) and (negative 2, negative 5) and an asymptote that passes through (negative 2, negative 1) and (8, negative 5), and branches that pass through the vertices (negative 2, negative 2) and (8, negative 2) and opens left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873645239\">\n<div data-type=\"problem\" id=\"fs-id1163873645242\">\n<p id=\"fs-id1163873645244\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5a8ca61cc89d18593795d01fc7efd2f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873809201\">In the following exercises, <span class=\"token\">\u24d0<\/span> write the equation in standard form and <span class=\"token\">\u24d1<\/span> graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870256623\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870256625\">\n<p id=\"fs-id1163870256627\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02e5c80dbadacb72d202e965ac193c12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#56;&#120;&#43;&#56;&#121;&#45;&#51;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"236\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873865095\">\n<p id=\"fs-id1163873865097\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d392ff2a86b81fe48c0c67dcba15c5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163873783464\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 1) an asymptote that passes through (3, 4) and (negative 1, negative 2) and an asymptote that passes through (negative 1, 4) and (3, negative 2), and branches that pass through the vertices (negative 1, 1) and (3, 1) and opens left and right.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_334_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 1) an asymptote that passes through (3, 4) and (negative 1, negative 2) and an asymptote that passes through (negative 1, 4) and (3, negative 2), and branches that pass through the vertices (negative 1, 1) and (3, 1) and opens left and right.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873799258\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873799260\">\n<p id=\"fs-id1163873799262\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f75b25364400215c8ea243d39420719d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#120;&#45;&#50;&#52;&#121;&#45;&#51;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"252\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873866311\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873866313\">\n<p id=\"fs-id1163873866315\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16a7b816eaf07bb291c9fa1e46587627_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#121;&#43;&#50;&#120;&#45;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"200\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873633020\">\n<p id=\"fs-id1163873633022\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef8af450b6958dcb4e9e26d44d0772b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"144\" style=\"vertical-align: -6px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163870407168\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 2) an asymptote that passes through (4, 5) and (negative 2, negative 1) and an asymptote that passes through (negative 2, 5) and (4, negative 1), and branches that pass through the vertices (1, 5) and (1, negative 1) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_336_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (1, 2) an asymptote that passes through (4, 5) and (negative 2, negative 1) and an asymptote that passes through (negative 2, 5) and (4, negative 1), and branches that pass through the vertices (1, 5) and (1, negative 1) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873919210\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873906647\">\n<p id=\"fs-id1163873906649\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ecb2a38e45e5c4d35216e677861fe32_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#52;&#121;&#43;&#57;&#54;&#120;&#45;&#49;&#55;&#50;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"262\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873819963\">\n<div data-type=\"problem\" id=\"fs-id1163873645897\">\n<p id=\"fs-id1163873645899\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61cbf905ff6d73fc343856fea616850b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#121;&#45;&#52;&#120;&#45;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873648568\">\n<p id=\"fs-id1163873648570\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ba3cf94a810b71759377f747f1f7a46f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><\/p>\n<p><span data-type=\"media\" id=\"fs-id1163870487208\" data-alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 2, negative 1) an asymptote that passes through (1, 0) and (negative 5, negative 2) and an asymptote that passes through (3, 0) and (1, negative 2), and branches that pass through the vertices (negative 2, 0) and (negative 2, negative 2) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_338_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x-axis and y-axis that both run in the negative and positive directions with the center (negative 2, negative 1) an asymptote that passes through (1, 0) and (negative 5, negative 2) and an asymptote that passes through (3, 0) and (1, negative 2), and branches that pass through the vertices (negative 2, 0) and (negative 2, negative 2) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1163870366282\"><strong data-effect=\"bold\">Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola<\/strong><\/p>\n<p id=\"fs-id1163870516339\">In the following exercises, identify the type of graph.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870516342\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163870516344\">\n<p id=\"fs-id1163870516346\"><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d431e52390c32e1c274db2382f7d57dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-61cbf905ff6d73fc343856fea616850b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#49;&#56;&#121;&#45;&#52;&#120;&#45;&#52;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"218\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28835fbf027b5c8f2a7d79d95d3af19a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#53;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#50;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc73afbac658af3f7fb1ab28608a4ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#48;&#121;&#45;&#55;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"209\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873796654\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1163873782194\">\n<p id=\"fs-id1163873782196\"><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07126c4db1199212ce3f552f3b5a7295_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#45;&#50;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#49;&#50;&#121;&#45;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"162\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4bee4195382ea0bf825e615ffd3b49da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"89\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f75b25364400215c8ea243d39420719d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#54;&#52;&#120;&#45;&#50;&#52;&#121;&#45;&#51;&#54;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"252\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d3<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a2b8725d10a46632dccb6da6b97c8bd0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#51;&#54;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#53;&#55;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870619458\">\n<p id=\"fs-id1163870619460\"><span class=\"token\">\u24d0<\/span> parabola <span class=\"token\">\u24d1<\/span> circle <span class=\"token\">\u24d2<\/span> hyperbola <span class=\"token\">\u24d3<\/span> ellipse<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1163873648694\"><strong data-effect=\"bold\">Mixed Practice<\/strong><\/p>\n<p id=\"fs-id1163870259320\">In the following exercises, graph each equation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1163870259323\">\n<div data-type=\"problem\" id=\"fs-id1163870259326\">\n<p id=\"fs-id1163870259328\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4663ea56d6309fad58f826b07848c1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#49;&#54;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"144\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873662392\">\n<div data-type=\"problem\" id=\"fs-id1163873869471\">\n<p id=\"fs-id1163873869473\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc73afbac658af3f7fb1ab28608a4ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#120;&#43;&#49;&#48;&#121;&#45;&#55;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"209\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873935276\"><span data-type=\"media\" id=\"fs-id1163873935279\" data-alt=\"The graph shows the x y coordinate plane with a circle whose center is (2, negative 5) and whose radius is 6 units.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_340_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x y coordinate plane with a circle whose center is (2, negative 5) and whose radius is 6 units.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873855097\">\n<div data-type=\"problem\" id=\"fs-id1163873855100\">\n<p id=\"fs-id1163873795847\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-542d25221fcec144f611df8e3793b8d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870644270\">\n<div data-type=\"problem\" id=\"fs-id1163870644272\">\n<p id=\"fs-id1163870644274\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-caf7f5d1e1008ae2c5409d331d8a3043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#50;&#53;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873557315\"><span data-type=\"media\" id=\"fs-id1163873811845\" data-alt=\"The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 5) and co-vertices are (plus or minus 3, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_342_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 5) and co-vertices are (plus or minus 3, 0).\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870292668\">\n<div data-type=\"problem\" id=\"fs-id1163870292670\">\n<p id=\"fs-id1163870292672\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1b5c256c22383ee489579b686a2171d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#43;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#121;&#45;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"177\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873804950\">\n<div data-type=\"problem\" id=\"fs-id1163873804952\">\n<p id=\"fs-id1163873655564\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-46a6f7e4332285926f9939c3b89836be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#45;&#52;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#43;&#53;&#52;&#120;&#43;&#56;&#121;&#43;&#52;&#49;&#61;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"236\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163870258892\"><span data-type=\"media\" id=\"fs-id1163870258895\" data-alt=\"The graph shows the x y coordinate plane with the center (1, 2) an asymptote that passes through (negative 2, 5) and (5, negative 1) and an asymptote that passes through (4, 5) and (2, 0), and branches that pass through the vertices (1, 5) and (negative 2, negative 1) and open up and down.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_344_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x y coordinate plane with the center (1, 2) an asymptote that passes through (negative 2, 5) and (5, negative 1) and an asymptote that passes through (4, 5) and (2, 0), and branches that pass through the vertices (1, 5) and (negative 2, negative 1) and open up and down.\" \/><\/span><\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873748917\">\n<div data-type=\"problem\" id=\"fs-id1163873748920\">\n<p id=\"fs-id1163873748922\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d431e52390c32e1c274db2382f7d57dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#8722;&#125;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#45;&#50;&#121;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870549277\">\n<div data-type=\"problem\" id=\"fs-id1163870549279\">\n<p id=\"fs-id1163870549282\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3d2704801c5f75b76e0a63301dcdc330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#43;&#57;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#61;&#49;&#52;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873654059\"><span data-type=\"media\" id=\"fs-id1163870548126\" data-alt=\"The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 4) and co-vertices are (plus or minus 3, 0).\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_346_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"The graph shows the x y coordinate plane with an ellipse whose major axis is vertical, vertices are (0, plus or minus 4) and co-vertices are (plus or minus 3, 0).\" \/><\/span><\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1163874054189\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1163874054196\">\n<div data-type=\"problem\" id=\"fs-id1163873654001\">\n<p id=\"fs-id1163873654003\">In your own words, define a hyperbola and write the equation of a hyperbola centered at the origin in standard form. Draw a sketch of the hyperbola labeling the center, vertices, and asymptotes.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163870693901\">\n<div data-type=\"problem\" id=\"fs-id1163870693903\">\n<p id=\"fs-id1163870693905\">Explain in your own words how to create and use the rectangle that helps graph a hyperbola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873641719\">\n<p id=\"fs-id1163873641721\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873641726\">\n<div data-type=\"problem\" id=\"fs-id1163873641728\">\n<p id=\"fs-id1163873641730\">Compare and contrast the graphs of the equations <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cfb13b9a265cbc4bb1cf71344e57106_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"88\" style=\"vertical-align: -6px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd16664f078a5072bd7911871f4d91e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#121;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#57;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#120;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"93\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1163873716520\">\n<div data-type=\"problem\" id=\"fs-id1163873716522\">\n<p id=\"fs-id1163873716524\">Explain in your own words, how to distinguish the equation of an ellipse with the equation of a hyperbola.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1163873861987\">\n<p id=\"fs-id1163873861989\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1163873861995\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1163874054311\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1163874054319\" data-alt=\"This table has four columns and four rows. The first row is a header and it labels each column, &#x201c;I can&#x2026;&#x201d;, &#x201c;Confidently,&#x201d; &#x201c;With some help,&#x201d; and &#x201c;No-I don&#x2019;t get it!&#x201d; In row 2, the I can was graph a hyperbola with center at (0, 0). In row 3, the I can was graph a hyperbola with a center at (h, k). In row 4, the I can was identify conic sections by their equations.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_11_04_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This table has four columns and four rows. The first row is a header and it labels each column, &#x201c;I can&#x2026;&#x201d;, &#x201c;Confidently,&#x201d; &#x201c;With some help,&#x201d; and &#x201c;No-I don&#x2019;t get it!&#x201d; In row 2, the I can was graph a hyperbola with center at (0, 0). In row 3, the I can was graph a hyperbola with a center at (h, k). In row 4, the I can was identify conic sections by their equations.\" \/><\/span><\/p>\n<p id=\"fs-id1163873892733\"><span class=\"token\">\u24d1<\/span> On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1163873673686\">\n<dt>hyperbola<\/dt>\n<dd id=\"fs-id1163873673691\">A hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15554","chapter","type-chapter","status-publish","hentry"],"part":15253,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15554","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15554\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/15253"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15554\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=15554"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=15554"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=15554"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=15554"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}