Graph Quadratic Functions Using Transformations

Lynn Marecek

Quadratic Equations and Functions

Graph Quadratic Functions Using Transformations

Learning Objectives

By the end of this section, you will be able to:

Graph quadratic functions of the form $f\left(x\right)={x}^{2}+k$
Graph quadratic functions of the form $f\left(x\right)={\left(x-h\right)}^{2}$
Graph quadratic functions of the form $f\left(x\right)=a{x}^{2}$
Graph quadratic functions using transformations
Find a quadratic function from its graph

Before you get started, take this readiness quiz.

Graph the function $f\left(x\right)={x}^{2}$ by plotting points.

If you missed this problem, review (Figure).
Factor completely: ${y}^{2}-14y+49.$

If you missed this problem, review (Figure).
Factor completely: $2{x}^{2}-16x+32.$

If you missed this problem, review (Figure).

Graph Quadratic Functions of the form $f\left(x\right)={x}^{2}+k$

In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of $f\left(x\right)={x}^{2}$ and ‘moving’ it according to information given in the function equation. We call this graphing quadratic functions using transformations.

In the first example, we will graph the quadratic function $f\left(x\right)={x}^{2}$ by plotting points. Then we will see what effect adding a constant, k, to the equation will have on the graph of the new function $f\left(x\right)={x}^{2}+k.$

Graph $f\left(x\right)={x}^{2},\phantom{\rule{0.2em}{0ex}}g\left(x\right)={x}^{2}+2,$ and $h\left(x\right)={x}^{2}-2$ on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Plotting points will help us see the effect of the constants on the basic $f\left(x\right)={x}^{2}$ graph. We fill in the chart for all three functions.

The g(x) values are two more than the f(x) values. Also, the h(x) values are two less than the f(x) values. Now we will graph all three functions on the same rectangular coordinate system.

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle is the graph of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top parabola has been moved up 2 units, and the bottom has been moved down 2 units.

The graph of $g\left(x\right)={x}^{2}+2$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted up 2 units.

The graph of $h\left(x\right)={x}^{2}-2$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted down 2 units.

ⓐ Graph $f\left(x\right)={x}^{2},\phantom{\rule{0.2em}{0ex}}g\left(x\right)={x}^{2}+1,$ and $h\left(x\right)={x}^{2}-1$ on the same rectangular coordinate system.

ⓑ Describe what effect adding a constant to the function has on the basic parabola.

ⓐ

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle graph is of f of x equals x squared has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 1 unit, and the bottom has been moved down 1 unit.

ⓑ The graph of $g\left(x\right)={x}^{2}+1$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted up 1 unit. The graph of $h\left(x\right)={x}^{2}-1$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted down 1 unit.

ⓐ Graph $f\left(x\right)={x}^{2},\phantom{\rule{0.2em}{0ex}}g\left(x\right)={x}^{2}+6,$ and $h\left(x\right)={x}^{2}-6$ on the same rectangular coordinate system.

ⓑ Describe what effect adding a constant to the function has on the basic parabola.

ⓐ

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 6 units, and the bottom has been moved down 6 units.

ⓑ The graph of $h\left(x\right)={x}^{2}+6$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted up 6 units. The graph of $h\left(x\right)={x}^{2}-6$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted down 6 units.

The last example shows us that to graph a quadratic function of the form $f\left(x\right)={x}^{2}+k,$ we take the basic parabola graph of $f\left(x\right)={x}^{2}$ and vertically shift it up $\left(k>0\right)$ or shift it down $\left(k<0\right)$ .

This transformation is called a vertical shift.

Graph a Quadratic Function of the form $f\left(x\right)={x}^{2}+k$ Using a Vertical Shift

The graph of $f\left(x\right)={x}^{2}+k$ shifts the graph of $f\left(x\right)={x}^{2}$ vertically k units.

If k > 0, shift the parabola vertically up k units.
If k < 0, shift the parabola vertically down $|k|$ units.

Now that we have seen the effect of the constant, k, it is easy to graph functions of the form $f\left(x\right)={x}^{2}+k.$ We just start with the basic parabola of $f\left(x\right)={x}^{2}$ and then shift it up or down.

It may be helpful to practice sketching $f\left(x\right)={x}^{2}$ quickly. We know the values and can sketch the graph from there.

This figure shows an upward-opening parabola on the x y-coordinate plane, with vertex (0, 0). Other points on the curve are located at (negative 4, 16), (negative 3, 9), (negative 2, 4), (negative 1, 1), (1, 1), (2, 4), (3, 9), and (4, 16).

Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.

Graph $f\left(x\right)={x}^{2}-3$ using a vertical shift.

We first draw the graph of $f\left(x\right)={x}^{2}$ on the grid.
Determine $k$ .

Shift the graph $f\left(x\right)={x}^{2}$ down 3.

Graph $f\left(x\right)={x}^{2}-5$ using a vertical shift.

This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The top curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The bottom curve has been moved down 5 units.

Graph $f\left(x\right)={x}^{2}+7$ using a vertical shift.

This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The bottom curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 7 units.

Graph Quadratic Functions of the form $f\left(x\right)={\left(x-h\right)}^{2}$

In the first example, we graphed the quadratic function $f\left(x\right)={x}^{2}$ by plotting points and then saw the effect of adding a constant k to the function had on the resulting graph of the new function $f\left(x\right)={x}^{2}+k.$

We will now explore the effect of subtracting a constant, h, from x has on the resulting graph of the new function $f\left(x\right)={\left(x-h\right)}^{2}.$

Graph $f\left(x\right)={x}^{2},\phantom{\rule{0.5em}{0ex}}g\left(x\right)={\left(x-1\right)}^{2},$ and $h\left(x\right)={\left(x+1\right)}^{2}$ on the same rectangular coordinate system. Describe what effect adding a constant to the function has on the basic parabola.

Plotting points will help us see the effect of the constants on the basic $f\left(x\right)={x}^{2}$ graph. We fill in the chart for all three functions.

The g(x) values and the h(x) values share the common numbers 0, 1, 4, 9, and 16, but are shifted.

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 1 unit, and the right curve has been moved to the right 1 unit.

ⓐ Graph $f\left(x\right)={x}^{2},\phantom{\rule{0.5em}{0ex}}g\left(x\right)={\left(x+2\right)}^{2},$ and $h\left(x\right)={\left(x-2\right)}^{2}$ on the same rectangular coordinate system.

ⓑ Describe what effect adding a constant to the function has on the basic parabola.

ⓐ

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 2 units, and the right curve has been moved to the right 2 units.

ⓑ The graph of $g\left(x\right)={\left(x+2\right)}^{2}$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted left 2 units. The graph of $h\left(x\right)={\left(x-2\right)}^{2}$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shift right 2 units.

ⓐ Graph $f\left(x\right)={x}^{2},\phantom{\rule{0.5em}{0ex}}g\left(x\right)={x}^{2}+5,$ and $h\left(x\right)={x}^{2}-5$ on the same rectangular coordinate system.

ⓑ Describe what effect adding a constant to the function has on the basic parabola.

ⓐ

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 5 units, and the right curve has been moved to the right 5 units.

ⓑ The graph of $g\left(x\right)={\left(x+5\right)}^{2}$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted left 5 units. The graph of $h\left(x\right)={\left(x-5\right)}^{2}$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted right 5 units.

The last example shows us that to graph a quadratic function of the form $f\left(x\right)={\left(x-h\right)}^{2},$ we take the basic parabola graph of $f\left(x\right)={x}^{2}$ and shift it left (h > 0) or shift it right (h < 0).

This transformation is called a horizontal shift.

Graph a Quadratic Function of the form $f\left(x\right)={\left(x-h\right)}^{2}$ Using a Horizontal Shift

The graph of $f\left(x\right)={\left(x-h\right)}^{2}$ shifts the graph of $f\left(x\right)={x}^{2}$ horizontally $h$ units.

If h > 0, shift the parabola horizontally left h units.
If h < 0, shift the parabola horizontally right $|h|$ units.

Now that we have seen the effect of the constant, h, it is easy to graph functions of the form $f\left(x\right)={\left(x-h\right)}^{2}.$ We just start with the basic parabola of $f\left(x\right)={x}^{2}$ and then shift it left or right.

The next example will require a horizontal shift.

Graph $f\left(x\right)={\left(x-6\right)}^{2}$ using a horizontal shift.

We first draw the graph of $f\left(x\right)={x}^{2}$ on the grid.
Determine h.

Shift the graph $f\left(x\right)={x}^{2}$ to the right 6 units.

Graph $f\left(x\right)={\left(x-4\right)}^{2}$ using a horizontal shift.

This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The left curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The right curve has been moved right 4 units.

Graph $f\left(x\right)={\left(x+6\right)}^{2}$ using a horizontal shift.

This figure shows 2 upward-opening parabolas on the x y-coordinate plane. The right curve is the graph of f of x equals x squared which has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The left curve has been moved to the left 6 units.

Now that we know the effect of the constants h and k, we will graph a quadratic function of the form $f\left(x\right)={\left(x-h\right)}^{2}+k$ by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.

Graph $f\left(x\right)={\left(x+1\right)}^{2}-2$ using transformations.

This function will involve two transformations and we need a plan.

Let’s first identify the constants h, k.

F of x equals the quantity x plush 1 squared minus 2 is given on the top line with f of x equals the quanitity x minus h squared minis k on the second line. The given equation was changed to f of x equals the quantity of x minus negative 1 squared plush negative 2 on the third line. The final line says h equals negative 1 and k equals negative 2.

The h constant gives us a horizontal shift and the k gives us a vertical shift.

F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared with an arrow coming from it pointing to f of x equals the quantity x plus 1 squared minus 2. The next lines say h equals negative 1 which means shift left 1 unit and k equals negative 2 which means shift down 2 units.

We first draw the graph of $f\left(x\right)={x}^{2}$ on the grid.

The figure says on the first line that the graph of f of x equals the quantity x plus 1 squared is the same as the graph of f of x equals x squared but shifted left 1 unit. The second line states that the graph of f of x equals the quantity x plus 1 squared minus 2 is the same as the graph of f of x equals the quantity x plus 1 squared but shifted down 2 units.

Graph $f\left(x\right)={\left(x+2\right)}^{2}-3$ using transformations.

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). Then, the original function is moved 2 units to the left to produce f of x equals the quantity of x plus 2 squared. The final curve is produced by moving down 3 units to produce f of x equals the quantity of x plus 2 squared minus 3.

Graph $f\left(x\right)={\left(x-3\right)}^{2}+1$ using transformations.

Graph Quadratic Functions of the Form $f\left(x\right)=a{x}^{2}$

So far we graphed the quadratic function $f\left(x\right)={x}^{2}$ and then saw the effect of including a constant h or k in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient a on the resulting graph of the new function $f\left(x\right)=a{x}^{2}.$

If we graph these functions, we can see the effect of the constant a, assuming a > 0.

To graph a function with constant a it is easiest to choose a few points on $f\left(x\right)={x}^{2}$ and multiply the y-values by a.

Graph of a Quadratic Function of the form $f\left(x\right)=a{x}^{2}$

The coefficient a in the function $f\left(x\right)=a{x}^{2}$ affects the graph of $f\left(x\right)={x}^{2}$ by stretching or compressing it.

If $0<|a|<1,$ the graph of $f\left(x\right)=a{x}^{2}$ will be “wider” than the graph of $f\left(x\right)={x}^{2}.$
If $|a|>1$ , the graph of $f\left(x\right)=a{x}^{2}$ will be “skinnier” than the graph of $f\left(x\right)={x}^{2}.$

Graph $f\left(x\right)=3{x}^{2}.$

We will graph the functions $f\left(x\right)={x}^{2}$ and $g\left(x\right)=3{x}^{2}$ on the same grid. We will choose a few points on $f\left(x\right)={x}^{2}$ and then multiply the y-values by 3 to get the points for $g\left(x\right)=3{x}^{2}.$

Graph $f\left(x\right)=-3{x}^{2}.$

Graph $f\left(x\right)=2{x}^{2}.$

This figure shows 2 upward-opening parabolas on the x y-coordinate plane. One is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The slimmer curve of f of x equals 2 times x square has a vertex at (0,0) and other points of (negative 1, one-half) and (1, one-half).

Graph Quadratic Functions Using Transformations

We have learned how the constants a, h, and k in the functions, $f\left(x\right)={x}^{2}+k,\phantom{\rule{0.2em}{0ex}}f\left(x\right)={\left(x-h\right)}^{2},$ and $f\left(x\right)=a{x}^{2}$ affect their graphs. We can now put this together and graph quadratic functions $f\left(x\right)=a{x}^{2}+bx+c$ by first putting them into the form $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ by completing the square. This form is sometimes known as the vertex form or standard form.

We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.

When we complete the square in a function with a coefficient of x² that is not one, we have to factor that coefficient from just the x-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the x-terms.

Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.

Rewrite $f\left(x\right)=-3{x}^{2}-6x-1$ in the $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form by completing the square.


Separate the x terms from the constant.
Factor the coefficient of ${x}^{2}$ , $-3$ .
Prepare to complete the square.
Take half of 2 and then square it to complete the square. ${\left(\frac{1}{2}·2\right)}^{2}=1$
The constant 1 completes the square in the parentheses, but the parentheses is multiplied by $-3$ . So we are really adding $-3$ We must then add 3 to not change the value of the function.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form.

Rewrite $f\left(x\right)=-4{x}^{2}-8x+1$ in the $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form by completing the square.

$f\left(x\right)=-4{\left(x+1\right)}^{2}+5$

Rewrite $f\left(x\right)=2{x}^{2}-8x+3$ in the $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form by completing the square.

$f\left(x\right)=2{\left(x-2\right)}^{2}-5$

Once we put the function into the $f\left(x\right)={\left(x-h\right)}^{2}+k$ form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.

Graph $f\left(x\right)={x}^{2}+6x+5$ by using transformations.

Step 1. Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ vertex form by completing the square.


Separate the x terms from the constant.
Take half of 6 and then square it to complete the square. ${\left(\frac{1}{2}·6\right)}^{2}=9$
We both add 9 and subtract 9 to not change the value of the function.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the $f\left(x\right)={\left(x-h\right)}^{2}+k$ form.

Step 2: Graph the function using transformations.

Looking at the h, k values, we see the graph will take the graph of $f\left(x\right)={x}^{2}$ and shift it to the left 3 units and down 4 units.

F of x equals x squared is given with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared with an arrow coming from it pointing to f of x equals the quantity x plus 3 squared minus 4. The next lines say h equals negative 3 which means shift left 3 unit and k equals negative 4 which means shift down 4 units

We first draw the graph of $f\left(x\right)={x}^{2}$ on the grid.

To graph f of x equals the quantity x plus 3 squared, shift the graph of f of x equals x squares to the left 3 units. To graph f of x equals the quantity x plus 3 squared minus 4, shift the graph the quantity x plus 3 squared down 4 units.

Graph $f\left(x\right)={x}^{2}+2x-3$ by using transformations.

Graph $f\left(x\right)={x}^{2}-8x+12$ by using transformations.

We list the steps to take to graph a quadratic function using transformations here.

Graph a quadratic function using transformations.

Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form by completing the square.
Graph the function using transformations.

Graph $f\left(x\right)=-2{x}^{2}-4x+2$ by using transformations.

Step 1. Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ vertex form by completing the square.


Separate the x terms from the constant.
We need the coefficient of ${x}^{2}$ to be one. We factor $-2$ from the x-terms.
Take half of 2 and then square it to complete the square. ${\left(\frac{1}{2}·2\right)}^{2}=1$
We add 1 to complete the square in the parentheses, but the parentheses is multiplied by $-2$ . Se we are really adding $-2$ . To not change the value of the function we add 2.
Rewrite the trinomial as a square and subtract the constants.
The function is now in the $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form.

Step 2. Graph the function using transformations.

We first draw the graph of $f\left(x\right)={x}^{2}$ on the grid.

Graph $f\left(x\right)=-3{x}^{2}+12x-4$ by using transformations.

This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (2,8) and other points of (1,5) and (3,5).

Graph $f\left(x\right)=-2{x}^{2}+12x-9$ by using transformations.

This figure shows a downward-opening parabola on the x y-coordinate plane with a vertex of (3, 9) and other points of (1, 1) and (5, 1).

Now that we have completed the square to put a quadratic function into $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form, we can also use this technique to graph the function using its properties as in the previous section.

If we look back at the last few examples, we see that the vertex is related to the constants h and k.

In each case, the vertex is (h, k). Also the axis of symmetry is the line x = h.

We rewrite our steps for graphing a quadratic function using properties for when the function is in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form.

Graph a quadratic function in the form $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ using properties.

Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form.
Determine whether the parabola opens upward, a > 0, or downward, a < 0.
Find the axis of symmetry, x = h.
Find the vertex, (h, k).
Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
Find the x-intercepts.
Graph the parabola.

ⓐ Rewrite $f\left(x\right)=2{x}^{2}+4x+5$ in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form and ⓑ graph the function using properties.

Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form by completing the square.	$f\left(x\right)=2{x}^{2}+4x+5$
	$f\left(x\right)=2\left({x}^{2}+2x\right)+5$
	$f\left(x\right)=2\left({x}^{2}+2x+1\right)+5-2$
	$f\left(x\right)=2{\left(x+1\right)}^{2}+3$
Identify the constants $a,h,k.$	$\phantom{\rule{1.2em}{0ex}}a=2\phantom{\rule{1em}{0ex}}h=-1\phantom{\rule{1em}{0ex}}k=3$
Since $a=2$ , the parabola opens upward.
The axis of symmetry is $x=h$ .	The axis of symmetry is $x=-1$ .
The vertex is $\left(h,k\right)$ .	The vertex is $\left(-1,3\right)$ .
Find the y-intercept by finding $f\left(0\right)$ .	$f\left(0\right)=2\cdot {0}^{2}+4\cdot 0+5$
	$f\left(0\right)=5$
	y-intercept $\left(0,5\right)$
Find the point symmetric to $\left(0,5\right)$ across the axis of symmetry.	$\left(-2,\phantom{\rule{0.2em}{0ex}}5\right)$
Find the x-intercepts.	The discriminant negative, so there are no x-intercepts. Graph the parabola.

ⓐ Rewrite $f\left(x\right)=3{x}^{2}-6x+5$ in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form and ⓑ graph the function using properties.

ⓐ $f\left(x\right)=3{\left(x-1\right)}^{2}+2$

ⓑ

The graph shown is an upward facing parabola with vertex (1, 2) and y-intercept (0, 5). The axis of symmetry is shown, x equals 1.

ⓐ Rewrite $f\left(x\right)=-2{x}^{2}+8x-7$ in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form and ⓑ graph the function using properties.

ⓐ $f\left(x\right)=-2{\left(x-2\right)}^{2}+1$

ⓑ

The graph shown is a downward facing parabola with vertex (2, 1) and x-intercepts (1, 0) and (3, 0). The axis of symmetry is shown, x equals 2.

Find a Quadratic Function from its Graph

So far we have started with a function and then found its graph.

Now we are going to reverse the process. Starting with the graph, we will find the function.

Determine the quadratic function whose graph is shown.

The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0, 7).

$\begin{array}{cccccc}\text{Since it is quadratic, we start with the}\hfill & & & & & \\ f\left(x\right)=a{\left(x-h\right)}^{2}+k\phantom{\rule{0.2em}{0ex}}\text{form.}\hfill & & & & & \\ \text{The vertex,}\phantom{\rule{0.2em}{0ex}}\left(h,k\right),\phantom{\rule{0.2em}{0ex}}\text{is}\phantom{\rule{0.2em}{0ex}}\left(-2,-1\right)\phantom{\rule{0.2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}h=-2\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}k=-1.\hfill & & & \hfill \phantom{\rule{3em}{0ex}}f\left(x\right)& =\hfill & a{\left(x-\left(-2\right)\right)}^{2}-1\hfill \\ \text{To find}\phantom{\rule{0.2em}{0ex}}a\text{, we use the}\phantom{\rule{0.2em}{0ex}}y\text{-intercept,}\phantom{\rule{0.2em}{0ex}}\left(0,7\right).\hfill & & & & & \\ \text{So}\phantom{\rule{0.2em}{0ex}}f\left(0\right)=7.\hfill & & & \hfill \phantom{\rule{3em}{0ex}}7& =\hfill & a{\left(0+2\right)}^{2}-1\hfill \\ \text{Solve for}\phantom{\rule{0.2em}{0ex}}a.\hfill & & & \hfill \phantom{\rule{3em}{0ex}}7& =\hfill & 4a-1\hfill \\ & & & \hfill \phantom{\rule{3em}{0ex}}8& =\hfill & 4a\hfill \\ & & & \hfill \phantom{\rule{3em}{0ex}}2& =\hfill & a\hfill \\ \text{Write the function.}\hfill & & & \hfill \phantom{\rule{3em}{0ex}}f\left(x\right)& =\hfill & a{\left(x-h\right)}^{2}+k\hfill \\ \text{Substitute in}\phantom{\rule{0.2em}{0ex}}h=-2,k=-1\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}a=2.\hfill & & & \hfill \phantom{\rule{3em}{0ex}}f\left(x\right)& =\hfill & 2{\left(x+2\right)}^{2}-1\hfill \end{array}$

Write the quadratic function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form whose graph is shown.

The graph shown is an upward facing parabola with vertex (3, negative 4) and y-intercept (0, 5).

$f\left(x\right)={\left(x-3\right)}^{2}-4$

Determine the quadratic function whose graph is shown.

The graph shown is an upward facing parabola with vertex (negative 3, negative 1) and y-intercept (0, 8).

$f\left(x\right)={\left(x+3\right)}^{2}-1$

Access these online resources for additional instruction and practice with graphing quadratic functions using transformations.

Key Concepts

Graph a Quadratic Function of the form Using a Vertical Shift
- The graph of shifts the graph of vertically k units.
  - If k > 0, shift the parabola vertically up k units.
  - If k < 0, shift the parabola vertically down $|k|$ units.
Graph a Quadratic Function of the form Using a Horizontal Shift
- The graph of shifts the graph of horizontally h units.
  - If h > 0, shift the parabola horizontally left h units.
  - If h < 0, shift the parabola horizontally right $|h|$ units.
Graph of a Quadratic Function of the form
- The coefficient a in the function $f\left(x\right)=a{x}^{2}$ affects the graph of $f\left(x\right)={x}^{2}$ by stretching or compressing it.
  
  If $0<|a|<1,$ then the graph of $f\left(x\right)=a{x}^{2}$ will be “wider” than the graph of $f\left(x\right)={x}^{2}.$
  
  If $|a|>1,$ then the graph of $f\left(x\right)=a{x}^{2}$ will be “skinnier” than the graph of $f\left(x\right)={x}^{2}.$
How to graph a quadratic function using transformations
1. Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form by completing the square.
2. Graph the function using transformations.
Graph a quadratic function in the vertex form using properties
1. Rewrite the function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form.
2. Determine whether the parabola opens upward, a > 0, or downward, a < 0.
3. Find the axis of symmetry, x = h.
4. Find the vertex, (h, k).
5. Find they-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.
6. Find the x-intercepts, if possible.
7. Graph the parabola.

Practice Makes Perfect

Graph Quadratic Functions of the form $f\left(x\right)={x}^{2}+k$

In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, k, to the function has on the basic parabola.

$f\left(x\right)={x}^{2},\phantom{\rule{0.2em}{0ex}}g\left(x\right)={x}^{2}+4,$ and $h\left(x\right)={x}^{2}-4.$

ⓐ

This figure shows 3 upward-opening parabolas on the x y-coordinate plane. The middle curve is the graph of f of x equals x squared and has a vertex of (0, 0). Other points on the curve are located at (negative 1, 1) and (1, 1). The top curve has been moved up 4 units, and the bottom has been moved down 4 units.

ⓑ The graph of $g\left(x\right)={x}^{2}+4$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted up 4 units. The graph of $h\left(x\right)={x}^{2}-4$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shift down 4 units.

$f\left(x\right)={x}^{2},\phantom{\rule{0.2em}{0ex}}g\left(x\right)={x}^{2}+7,$ and $h\left(x\right)={x}^{2}-7.$

In the following exercises, graph each function using a vertical shift.

$f\left(x\right)={x}^{2}+3$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 3) and other points (7, 2) and (7, negative 2).

$f\left(x\right)={x}^{2}-7$

$g\left(x\right)={x}^{2}+2$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 2) and other points (negative 2, 6) and (2, 6).

$g\left(x\right)={x}^{2}+5$

$h\left(x\right)={x}^{2}-4$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).

$h\left(x\right)={x}^{2}-5$

Graph Quadratic Functions of the form $f\left(x\right)={\left(x-h\right)}^{2}$

In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, $h$ , inside the parentheses has

$f\left(x\right)={x}^{2},\phantom{\rule{0.2em}{0ex}}g\left(x\right)={\left(x-3\right)}^{2},$ and $h\left(x\right)={\left(x+3\right)}^{2}.$

ⓐ

ⓑ The graph of $g\left(x\right)={\left(x-3\right)}^{2}$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted right 3 units. The graph of $h\left(x\right)={\left(x+3\right)}^{2}$ is the same as the graph of $f\left(x\right)={x}^{2}$ but shifted left 3 units.

$f\left(x\right)={x}^{2},\phantom{\rule{0.2em}{0ex}}g\left(x\right)={\left(x+4\right)}^{2},$ and $h\left(x\right)={\left(x-4\right)}^{2}.$

In the following exercises, graph each function using a horizontal shift.

$f\left(x\right)={\left(x-2\right)}^{2}$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (2, 0) and other points (0, 4) and (4, 4).

$f\left(x\right)={\left(x-1\right)}^{2}$

$f\left(x\right)={\left(x+5\right)}^{2}$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 5, 0) and other points (negative 7, 4) and (negative 3, 4).

$f\left(x\right)={\left(x+3\right)}^{2}$

$f\left(x\right)={\left(x-5\right)}^{2}$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (5, 0) and other points (3, 4) and (7, 4).

$f\left(x\right)={\left(x+2\right)}^{2}$

In the following exercises, graph each function using transformations.

$f\left(x\right)={\left(x+2\right)}^{2}+1$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, 1) and other points (negative 4, 5) and (0, 5).

$f\left(x\right)={\left(x+4\right)}^{2}+2$

$f\left(x\right)={\left(x-1\right)}^{2}+5$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (1, 5) and other points (negative 1, 9) and (3, 9).

$f\left(x\right)={\left(x-3\right)}^{2}+4$

$f\left(x\right)={\left(x+3\right)}^{2}-1$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 1) and other points (negative 4, 0) and (negative 2, 0).

$f\left(x\right)={\left(x+5\right)}^{2}-2$

$f\left(x\right)={\left(x-4\right)}^{2}-3$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (4, negative 2) and other points (3, negative 2) and (5, negative 2).

$f\left(x\right)={\left(x-6\right)}^{2}-2$

Graph Quadratic Functions of the form $f\left(x\right)=a{x}^{2}$

In the following exercises, graph each function.

$f\left(x\right)=-2{x}^{2}$

This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 2) and (1, negative 2).

$f\left(x\right)=4{x}^{2}$

$f\left(x\right)=-4{x}^{2}$

This figure shows a downward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 1, negative 4) and (1, negative 4).

$f\left(x\right)=\text{−}{x}^{2}$

$f\left(x\right)=\frac{1}{2}{x}^{2}$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (negative 2, 2) and (2, 2).

$f\left(x\right)=\frac{1}{3}{x}^{2}$

$f\left(x\right)=\frac{1}{4}{x}^{2}$

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (0, 0) and other points (2, 1) and (negative 2, 1).

$f\left(x\right)=-\frac{1}{2}{x}^{2}$

Graph Quadratic Functions Using Transformations

In the following exercises, rewrite each function in the $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form by completing the square.

$f\left(x\right)=-3{x}^{2}-12x-5$

$f\left(x\right)=-3{\left(x+2\right)}^{2}+7$

$f\left(x\right)=2{x}^{2}-12x+7$

$f\left(x\right)=3{x}^{2}+6x-1$

$f\left(x\right)=3{\left(x+1\right)}^{2}-4$

$f\left(x\right)=-4{x}^{2}-16x-9$

In the following exercises, ⓐ rewrite each function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form and ⓑ graph it by using transformations.

$f\left(x\right)={x}^{2}+6x+5$

ⓐ $f\left(x\right)={\left(x+3\right)}^{2}-4$

ⓑ

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 3, 3), y-intercept of (0, 5), and axis of symmetry shown at x equals negative 3.

$f\left(x\right)={x}^{2}+4x-12$

ⓐ $f\left(x\right)={\left(x+2\right)}^{2}-1$

ⓑ

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (negative 2, negative 1), y-intercept of (0, 3), and axis of symmetry shown at x equals negative 2.

$f\left(x\right)={x}^{2}-6x+8$

$f\left(x\right)={x}^{2}-6x+15$

ⓐ $f\left(x\right)={\left(x-3\right)}^{2}+6$

ⓑ

This figure shows an upward-opening parabolas on the x y-coordinate plane. It has a vertex of (3, 6), y-intercept of (0, 10), and axis of symmetry shown at x equals 3.

$f\left(x\right)={x}^{2}+8x+10$

$f\left(x\right)=\text{−}{x}^{2}+8x-16$

ⓐ $f\left(x\right)=\text{−}{\left(x-4\right)}^{2}+0$

ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0), y-intercept of (0, negative 16), and axis of symmetry shown at x equals 4.

$f\left(x\right)=\text{−}{x}^{2}+2x-7$

$f\left(x\right)=\text{−}{x}^{2}-4x+2$

ⓐ $f\left(x\right)=\text{−}{\left(x+2\right)}^{2}+6$

ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 2, 6), y-intercept of (0, 2), and axis of symmetry shown at x equals negative 2.

$f\left(x\right)=\text{−}{x}^{2}+4x-5$

$f\left(x\right)=5{x}^{2}-10x+8$

ⓐ $f\left(x\right)=5{\left(x-1\right)}^{2}+3$

ⓑ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, 3), y-intercept of (0, 8), and axis of symmetry shown at x equals 1.

$f\left(x\right)=3{x}^{2}+18x+20$

$f\left(x\right)=2{x}^{2}-4x+1$

ⓐ $f\left(x\right)=2{\left(x-1\right)}^{2}-1$

ⓑ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 1), y-intercept of (0, 1), and axis of symmetry shown at x equals 1.

$f\left(x\right)=3{x}^{2}-6x-1$

$f\left(x\right)=-2{x}^{2}+8x-10$

ⓐ $f\left(x\right)=-2{\left(x-2\right)}^{2}-2$

ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (2, negative 2), y-intercept of (0, negative 10), and axis of symmetry shown at x equals 2.

$f\left(x\right)=-3{x}^{2}+6x+1$

In the following exercises, ⓐ rewrite each function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form and ⓑ graph it using properties.

$f\left(x\right)=2{x}^{2}+4x+6$

ⓐ $f\left(x\right)=2{\left(x+1\right)}^{2}+4$

ⓑ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, 4), y-intercept of (0, 6), and axis of symmetry shown at x equals negative 1.

$f\left(x\right)=3{x}^{2}-12x+7$

$f\left(x\right)=\text{−}{x}^{2}+2x-4$

ⓐ $f\left(x\right)=\text{−}{\left(x-1\right)}^{2}-3$

ⓑ

This figure shows a downward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3), y-intercept of (0, negative 4), and axis of symmetry shown at x equals 1.

$f\left(x\right)=-2{x}^{2}-4x-5$

Matching

In the following exercises, match the graphs to one of the following functions: ⓐ $f\left(x\right)={x}^{2}+4$ ⓑ $f\left(x\right)={x}^{2}-4$ ⓒ $f\left(x\right)={\left(x+4\right)}^{2}$ ⓓ $f\left(x\right)={\left(x-4\right)}^{2}$ ⓔ $f\left(x\right)={\left(x+4\right)}^{2}-4$ ⓕ $f\left(x\right)={\left(x+4\right)}^{2}+4$ ⓖ $f\left(x\right)={\left(x-4\right)}^{2}-4$ ⓗ $f\left(x\right)={\left(x-4\right)}^{2}+4$

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 0) and other points (negative 4, 4) and (negative 2, 4).

ⓒ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, negative 4) and other points (negative 2, 0) and (2, 0).

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, negative 4) and other points (negative 4, 0) and (negative 2, 0).

ⓔ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 4, 4) and other points (negative 6, 8) and (negative 2, 8).

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 0) and other points (2, 4) and (2, 4).

ⓓ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (0, 4) and other points (negative 2, 8) and (2, 8).

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, negative 4) and other points (2,0) and (6,0).

ⓖ

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (4, 4) and other points (2,8) and (6,8).

Find a Quadratic Function from its Graph

In the following exercises, write the quadratic function in $f\left(x\right)=a{\left(x-h\right)}^{2}+k$ form whose graph is shown.

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 4).

$f\left(x\right)={\left(x+1\right)}^{2}-5$

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (2,4) and y-intercept (0, 8).

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (1, negative 3) and y-intercept (0, negative 1).

$f\left(x\right)=2{\left(x-1\right)}^{2}-3$

This figure shows an upward-opening parabola on the x y-coordinate plane. It has a vertex of (negative 1, negative 5) and y-intercept (0, negative 3).

Writing Exercise

Graph the quadratic function $f\left(x\right)={x}^{2}+4x+5$ first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?

Answers will vary.

Graph the quadratic function $f\left(x\right)=2{x}^{2}-4x-3$ first using the properties as we did in the last section and then graph it using transformations. Which method do you prefer? Why?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

License

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Learning Objectives

Graph Quadratic Functions of the form

Graph Quadratic Functions of the form

Graph Quadratic Functions of the Form

Graph Quadratic Functions Using Transformations

Find a Quadratic Function from its Graph

Key Concepts

Practice Makes Perfect

Writing Exercise

Self Check

License

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Graph Quadratic Functions of the form $f\left(x\right)={x}^{2}+k$

Graph Quadratic Functions of the form $f\left(x\right)={\left(x-h\right)}^{2}$

Graph Quadratic Functions of the Form $f\left(x\right)=a{x}^{2}$