Graphs of Functions

Lynn Marecek

Graphs and Functions

Graphs of Functions

Learning Objectives

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

Use the vertical line test
Identify graphs of basic functions
Read information from a graph of a function

Before you get started, take this readiness quiz.

Evaluate: ⓐ ${2}^{3}$ ⓑ ${3}^{2}.$

If you missed this problem, review (Figure).
Evaluate: ⓐ $|7|$ ⓑ $|-3|.$

If you missed this problem, review (Figure).
Evaluate: ⓐ $\sqrt{4}$ ⓑ $\sqrt{16}.$

If you missed this problem, review (Figure).

Use the Vertical Line Test

In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function.

An ordered pair $\left(x,y\right)$ is a solution of a linear equation, if the equation is a true statement when the x– and y-values of the ordered pair are substituted into the equation.

The graph of a linear equation is a straight line where every point on the line is a solution of the equation and every solution of this equation is a point on this line.

In (Figure), we can see that, in graph of the equation $y=2x-3,$ for every x-value there is only one y-value, as shown in the accompanying table.

plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled y equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label y equals2 x minus 3. The second row is a header row with the headers x, y, and (x, y). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5).

A relation is a function if every element of the domain has exactly one value in the range. So the relation defined by the equation $y=2x-3$ is a function.

If we look at the graph, each vertical dashed line only intersects the line at one point. This makes sense as in a function, for every x-value there is only one y-value.

If the vertical line hit the graph twice, the x-value would be mapped to two y-values, and so the graph would not represent a function.

This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.

Vertical Line Test

A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.

If any vertical line intersects the graph in more than one point, the graph does not represent a function.

Determine whether each graph is the graph of a function.

ⓐ Since any vertical line intersects the graph in at most one point, the graph is the graph of a function.

The figure has a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 2), (3, 0), and (6, negative 2). Three dashed vertical straight lines are drawn at x equalsnegative 5, x equalsnegative 3, and x equals3. Each line intersects the slanted line at exactly one point.

ⓑ One of the vertical lines shown on the graph, intersects it in two points. This graph does not represent a function.

Determine whether each graph is the graph of a function.

ⓐ yes ⓑ no

Determine whether each graph is the graph of a function.

ⓐ no ⓑ yes

Identify Graphs of Basic Functions

We used the equation $y=2x-3$ and its graph as we developed the vertical line test. We said that the relation defined by the equation $y=2x-3$ is a function.

We can write this as in function notation as $f\left(x\right)=2x-3.$ It still means the same thing. The graph of the function is the graph of all ordered pairs $\left(x,y\right)$ where $y=f\left(x\right).$ So we can write the ordered pairs as $\left(x,f\left(x\right)\right).$ It looks different but the graph will be the same.

Compare the graph of $y=2x-3$ previously shown in (Figure) with the graph of $f\left(x\right)=2x-3$ shown in (Figure). Nothing has changed but the notation.

This figure has a graph next to a table. The graph has a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 3), (1, negative 1), and (2, 1). The line is labeled f of x equals2 x minus 3. There are several vertical arrows that relate values on the x-axis to points on the line. The first arrow relates x equalsnegative 2 on the x-axis to the point (negative 2, negative 7) on the line. The second arrow relates x equalsnegative 1 on the x-axis to the point (negative 1, negative 5) on the line. The next arrow relates x equals0 on the x-axis to the point (0, negative 3) on the line. The next arrow relates x equals3 on the x-axis to the point (3, 3) on the line. The last arrow relates x equals4 on the x-axis to the point (4, 5) on the line. The table has 7 rows and 3 columns. The first row is a title row with the label f of x equals2 x minus 3. The second row is a header row with the headers x, f of x, and (x, f of x). The third row has the coordinates negative 2, negative 7, and (negative 2, negative 7). The fourth row has the coordinates negative 1, negative 5, and (negative 1, negative 5). The fifth row has the coordinates 0, negative 3, and (0, negative 3). The sixth row has the coordinates 3, 3, and (3, 3). The seventh row has the coordinates 4, 5, and (4, 5).

Graph of a Function

The graph of a function is the graph of all its ordered pairs, $\left(x,y\right)$ or using function notation, $\left(x,f\left(x\right)\right)$ where $y=f\left(x\right).$

$\begin{array}{cccc}\hfill f& & & \text{name of function}\hfill \\ \hfill x& & & x\text{-coordinate of the ordered pair}\hfill \\ \hfill f\left(x\right)& & & y\text{-coordinate of the ordered pair}\hfill \end{array}$

As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them.

Through our earlier work, we are familiar with the graphs of linear equations. The process we used to decide if $y=2x-3$ is a function would apply to all linear equations. All non-vertical linear equations are functions. Vertical lines are not functions as the x-value has infinitely many y-values.

We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is $y=mx+b.$ In function notation, this linear function becomes $f\left(x\right)=mx+b$ where m is the slope of the line and b is the y-intercept.

The domain is the set of all real numbers, and the range is also the set of all real numbers.

Linear Function

This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsm x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.

We will use the graphing techniques we used earlier, to graph the basic functions.

Graph: $f\left(x\right)=-2x-4.$

	$\phantom{\rule{4.5em}{0ex}}f\left(x\right)=-2x-4$
We recognize this as a linear function.
Find the slope and y-intercept.	$\phantom{\rule{5.5em}{0ex}}m=-2$ $\phantom{\rule{5.65em}{0ex}}b=-4$
Graph using the slope intercept.

Graph: $f\left(x\right)=-3x-1$

The figure has the graph of a linear function on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The line goes through the points (1, negative 4), (0, negative 1), and (negative 1, 2).

Graph: $f\left(x\right)=-4x-5$

The figure has the graph of a linear function on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The line goes through the points (negative 2, 3), (0, negative 5), and (negative 1, negative 1).

The next function whose graph we will look at is called the constant function and its equation is of the form $f\left(x\right)=b,$ where b is any real number. If we replace the $f\left(x\right)$ with y, we get $y=b.$ We recognize this as the horizontal line whose y-intercept is b. The graph of the function $f\left(x\right)=b,$ is also the horizontal line whose y-intercept is b.

Notice that for any real number we put in the function, the function value will be b. This tells us the range has only one value, b.

Constant Function

This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsb”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.

Graph: $f\left(x\right)=4.$

	$f\left(x\right)=4$
We recognize this as a constant function.
The graph will be a horizontal line through $\left(0,4\right).$

Graph: $f\left(x\right)=-2.$

The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, negative 2), (1, negative 2), and (2, negative 2).

Graph: $f\left(x\right)=3.$

The figure has the graph of a constant function on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 3), (1, 3), and (2, 3).

The identity function, $f\left(x\right)=x$ is a special case of the linear function. If we write it in linear function form, $f\left(x\right)=1x+0,$ we see the slope is 1 and the y-intercept is 0.

Identity Function

This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.

The next function we will look at is not a linear function. So the graph will not be a line. The only method we have to graph this function is point plotting. Because this is an unfamiliar function, we make sure to choose several positive and negative values as well as 0 for our x-values.

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

We choose x-values. We substitute them in and then create a chart as shown.

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

This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4).

$f\left(x\right)=\text{−}{x}^{2}$

This figure has a graph next to a table. In the graph there is a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, negative 4), (negative 1, negative 1), (0, 0), (1, negative 1), and (2, negative 4).

Looking at the result in (Figure), we can summarize the features of the square function. We call this graph a parabola. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.

The range is not all real numbers. Notice the graph consists of values of y never go below zero. This makes sense as the square of any number cannot be negative. So, the range of the square function is all non-negative real numbers.

Square Function

This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.

The next function we will look at is also not a linear function so the graph will not be a line. Again we will use point plotting, and make sure to choose several positive and negative values as well as 0 for our x-values.

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

We choose x-values. We substitute them in and then create a chart.

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

This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).

Graph: $f\left(x\right)=\text{−}{x}^{3}.$

This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, 8), (negative 1, 1), (0, 0), (1, negative 1), and (2, negative 8).

Looking at the result in (Figure), we can summarize the features of the cube function. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.

The range is all real numbers. This makes sense as the cube of any non-zero number can be positive or negative. So, the range of the cube function is all real numbers.

Cube Function

The next function we will look at does not square or cube the input values, but rather takes the square root of those values.

Let’s graph the function $f\left(x\right)=\sqrt{x}$ and then summarize the features of the function. Remember, we can only take the square root of non-negative real numbers, so our domain will be the non-negative real numbers.

$f\left(x\right)=\sqrt{x}$

We choose x-values. Since we will be taking the square root, we choose numbers that are perfect squares, to make our work easier. We substitute them in and then create a chart.

Graph: $f\left(x\right)=\sqrt{x}.$

This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1), (4, 2), and (9, 3).

Graph: $f\left(x\right)=\text{−}\sqrt{x}.$

This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from negative 10 to 0. The curved half-line starts at the point (0, 0) and then goes down and to the right. The curved half line goes through the points (1, negative 1), (4, negative 2), and (9, negative 3).

Square Root Function

This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph are the following: “f of x equalssquare root of x”, “Domain: [0, infinity)”, and “Range: [0, infinity)”.

Our last basic function is the absolute value function, $f\left(x\right)=|x|.$ Keep in mind that the absolute value of a number is its distance from zero. Since we never measure distance as a negative number, we will never get a negative number in the range.

Graph: $f\left(x\right)=|x|.$

We choose x-values. We substitute them in and then create a chart.

Graph: $f\left(x\right)=|x|.$

This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3).

Graph: $f\left(x\right)=\text{−}|x|.$

This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 8 to 4. The v-shaped line goes through the points (negative 3, negative 3), (negative 2, negative 2), (negative 1, negative 1), (0, 0), (1, negative 1), (2, negative 2), and (3, negative 3).

Absolute Value Function

Read Information from a Graph of a Function

In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data.

We will start by reading the domain and range of a function from its graph.

Remember the domain is the set of all the x-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of x that have a corresponding value on the graph. Follow the value x up or down vertically. If you hit the graph of the function then x is in the domain.

Remember the range is the set of all the y-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of y that have a corresponding value on the graph. Follow the value y left or right horizontally. If you hit the graph of the function then y is in the range.

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line segment goes through the points (negative 3, negative 1), (1.5, 3), and (3, 1). The interval [negative 3, 3] is marked on the horizontal axis. The interval [negative 1, 3] is marked on the vertical axis.

To find the domain we look at the graph and find all the values of x that correspond to a point on the graph. The domain is highlighted in red on the graph. The domain is $\left[-3,3\right].$

To find the range we look at the graph and find all the values of y that correspond to a point on the graph. The range is highlighted in blue on the graph. The range is $\left[-1,3\right].$

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 5, negative 4), (0, negative 3), and (1, 2). The interval [negative 5, 1] is marked on the horizontal axis. The interval [negative 4, 2] is marked on the vertical axis.

The domain is $\left[-5,1\right].$ The range is $\left[-4,2\right].$

Use the graph of the function to find its domain and range. Write the domain and range in interval notation.

This figure has a curved line segment graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 5. The y-axis runs from negative 6 to 4. The curved line segment goes through the points (negative 2, 1), (0, 3), and (4, negative 5). The interval [negative 2, 4] is marked on the horizontal axis. The interval [negative 5, 3] is marked on the vertical axis.

The domain is $\left[-2,4\right].$ The range is $\left[-5,3\right].$

We are now going to read information from the graph that you may see in future math classes.

Use the graph of the function to find the indicated values.

ⓐ Find: $f\left(0\right).$

ⓑ Find: $f\left(\frac{3}{2}\pi \right).$

ⓓ Find the values for x when $f\left(x\right)=0.$

ⓔ Find the x-intercepts.

ⓕ Find the y-intercepts.

ⓖ Find the domain. Write it in interval notation.

ⓗ Find the range. Write it in interval notation.

ⓐ When $x=0,$ the function crosses the y-axis at 0. So, $f\left(0\right)=0.$

ⓑ When $x=\frac{3}{2}\pi ,$ the y-value of the function is $-1.$ So, $f\left(\frac{3}{2}\pi \right)=-1.$

ⓓ The function is 0 at the points, $\left(-2\pi ,0\right),\left(\text{−}\pi ,0\right),\left(0,0\right),\left(\pi ,0\right),\left(2\pi ,0\right).$ The x-values when $f\left(x\right)=0$ are $-2\pi ,\text{−}\pi ,0,\pi ,2\pi .$

ⓔ The x-intercepts occur when $y=0.$ So the x-intercepts occur when $f\left(x\right)=0.$ The x-intercepts are $\left(-2\pi ,0\right),\left(\text{−}\pi ,0\right),\left(0,0\right),\left(\pi ,0\right),\left(2\pi ,0\right).$

ⓕ The y-intercepts occur when $x=0.$ So the y-intercepts occur at $f\left(0\right).$ The y-intercept is $\left(0,0\right).$

ⓖ This function has a value when x is from $-2\pi$ to $2\pi .$ Therefore, the domain in interval notation is $\left[-2\pi ,2\pi \right].$

ⓗ This function values, or y-values go from $-1$ to 1. Therefore, the range, in interval notation, is $\left[-1,1\right].$

Use the graph of the function to find the indicated values.

ⓐ Find: $f\left(0\right).$

ⓑ Find: $f\left(\frac{1}{2}\pi \right).$

ⓓ Find the values for x when $f\left(x\right)=0.$

ⓔ Find the x-intercepts.

ⓕ Find the y-intercepts.

ⓖ Find the domain. Write it in interval notation.

ⓗ Find the range. Write it in interval notation.

ⓐ $f\left(0\right)=0$ ⓑ $f=\left(\frac{\pi }{2}\right)=2$ ⓒ $f=\left(\frac{-3\pi }{2}\right)=2$ ⓓ $f\left(x\right)=0$ for $x=-2\pi ,\text{−}\pi ,0,\pi ,2\pi$ ⓔ $\left(-2\pi ,0\right),\left(\text{−}\pi ,0\right),\left(0,0\right),\left(\pi ,0\right),\left(2\pi ,0\right)$ ⓕ $\left(0,0\right)$ ⓖ $\left[-2\pi ,2\pi \right]$ ⓗ $\left[-2,2\right]$

Use the graph of the function to find the indicated values.

ⓐ Find: $f\left(0\right).$

ⓑ Find: $f\left(\pi \right).$

ⓓ Find the values for x when $f\left(x\right)=0.$

ⓔ Find the x-intercepts.

ⓕ Find the y-intercepts.

ⓖ Find the domain. Write it in interval notation.

ⓗ Find the range. Write it in interval notation.

ⓐ $f\left(0\right)=1$ ⓑ $f\left(\pi \right)=-1$ ⓒ $f\left(\text{−}\pi \right)=-1$ ⓓ $f\left(x\right)=0$ for $x=-\frac{3\pi }{2},-\frac{\pi }{2},\frac{\pi }{2},\frac{3\pi }{2}$ ⓔ $\left(-2\text{pi},0\right),\left(\text{−pi},0\right),\left(0,0\right),\left(\text{pi},0\right),\left(2\text{pi},0\right)$ ⓕ $\left(0,1\right)$ ⓖ $\left[-2\text{pi},2\text{pi}\right]$ ⓗ $\left[-1,1\right]$

Access this online resource for additional instruction and practice with graphs of functions.

Find Domain and Range

Key Concepts

Vertical Line Test
- A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
- If any vertical line intersects the graph in more than one point, the graph does not represent a function.
Graph of a Function
- The graph of a function is the graph of all its ordered pairs, $\left(x,y\right)$ or using function notation, $\left(x,f\left(x\right)\right)$ where $y=f\left(x\right).$
  
  $\begin{array}{cccc}\hfill f& & & \text{name of function}\hfill \\ \hfill x& & & x\text{-coordinate of the ordered pair}\hfill \\ \hfill f\left(x\right)& & & y\text{-coordinate of the ordered pair}\hfill \end{array}$
Linear Function
Constant Function
Identity Function
Square Function
Cube Function
Square Root Function
Absolute Value Function

Section Exercises

Practice Makes Perfect

Use the Vertical Line Test

In the following exercises, determine whether each graph is the graph of a function.

ⓐ

The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 3, 0), (3, 0), (0, negative 3), and (0, 3).

ⓑ

The figure has a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (1, 3), (0, 2), (1, 3), and (2, 6).

ⓐ no ⓑ yes

ⓐ

The figure has an s-shaped curved line graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The s-shaped curved line goes through the points (negative 1, 1), (0, 0), and (1, 1).

ⓑ

The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 4, 0), (4, 0), (0, negative 4), and (0, 4).

ⓐ

The figure has a parabola opening right graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (negative 2, 2), and (2, 2).

ⓑ

The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).

ⓐ no ⓑ yes

ⓐ

The figure has two curved lines graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line on the left goes through the points (negative 2, 0), (negative 4, 5), and (negative 4, negative 5). The curved line on the right goes through the points (2, 0), (4, 5), and (4, negative 5).

ⓑ

The figure has a sideways absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line bends at the point (0, 2) and goes to the right. The line goes through the points (1, 3), (2, 4), (1, 1), and (2, 0).

Identify Graphs of Basic Functions

In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.

$f\left(x\right)=3x+4$

ⓐ

The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, negative 2), (negative 1, 1), and (0, 4).

ⓑ D:(-∞,∞), R:(-∞,∞)

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

$f\left(x\right)=\text{−}x-2$

ⓐ

The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 0), (0, negative 2), and (2, negative 4).

ⓑ D:(-∞,∞), R:(-∞,∞)

$f\left(x\right)=-4x-3$

$f\left(x\right)=-2x+2$

ⓐ

The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 2), (negative 1, 0), and (0, negative 2).

ⓑ D:(-∞,∞), R:(-∞,∞)

$f\left(x\right)=-3x+3$

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

ⓐ

The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 0), (0, 1), and (2, 2).

ⓑ D:(-∞,∞), R:(-∞,∞)

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

$f\left(x\right)=5$

ⓐ

The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 2, 5), (negative 1, 5), and (0, 5).

ⓑ D:(-∞,∞), R:{5}

$f\left(x\right)=2$

$f\left(x\right)=-3$

ⓐ

The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (0, negative 3), (1, negative 3), and (2, negative 3).

ⓑ D:(-∞,∞), R: $\left\{-3\right\}$

$f\left(x\right)=-1$

$f\left(x\right)=2x$

ⓐ

The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (0, 0), (2, 4), and (negative 2, negative 4).

ⓑ D:(-∞,∞), R:(-∞,∞)

$f\left(x\right)=3x$

$f\left(x\right)=-2x$

ⓐ

The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 12 to 12. The y-axis runs from negative 12 to 12. The line goes through the points (0, 0), (1, negative 2), and (negative 1, 2).

ⓑ D:(-∞,∞), R:(-∞,∞)

$f\left(x\right)=-3x$

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

ⓐ

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 1, 3), (0, 0), and (1, 3). The lowest point on the graph is (0, 0).

ⓑ D:(-∞,∞), R:[0,∞)

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

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

ⓐ

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 10 to 2. The parabola goes through the points (negative 1, negative 3), (0, 0), and (1, negative 3). The highest point on the graph is (0, 0).

ⓑ (-∞,∞), R:(-∞,0]

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

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

ⓐ

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 4, 8), (negative 2, 2), (0, 0), (2, 2), and (4, 8). The lowest point on the graph is (0, 0).

ⓑ (-∞,∞), R:[-∞,0)

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

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

ⓐ

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 3), (negative 1, 0), (0, negative 1), (1, 0), and (2, 3). The lowest point on the graph is (0, negative 1).

ⓑ (-∞,∞), R:[ $-1,$ ∞)

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

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

ⓐ

The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 2), (0, 0), and (1, negative 2).

ⓑ D:(-∞,∞), R:(-∞,∞)

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

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

ⓐ

The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 1), (0, 2), and (1, 3).

ⓑ D:(-∞,∞), R:(-∞,∞)

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

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

ⓐ

The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (0, 0) and goes through the points (1, 2) and (4, 4).

ⓑ D:[0,∞), R:[0,∞)

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

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

ⓐ

The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (1, 0) and goes through the points (2, 1) and (5, 2).

ⓑ D:[1,∞), R:[0,∞)

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

$f\left(x\right)=3|x|$

ⓐ

The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 0). The line goes through the points (negative 1, 3) and (1, 3).

ⓑ D:[ $-1,$ ∞), R:[−∞,∞)

$f\left(x\right)=-2|x|$

$f\left(x\right)=|x|+1$

ⓐ

The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 1). The line goes through the points (negative 1, 2) and (1, 2).

ⓑ D:(-∞,∞), R:[1,∞)

$f\left(x\right)=|x|-1$

Read Information from a Graph of a Function

In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation.

The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 2 to 8. The y-axis runs from negative 2 to 8. The half-line starts at the point (2, 0) and goes through the points (3, 1) and (6, 2).

D: [2,∞), R: [0,∞)

The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 2 to 8. The y-axis runs from negative 2 to 10. The half-line starts at the point (negative 3, 0) and goes through the points (negative 2, 1) and (1, 2).

The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from 0 to 12. The vertex is at the point (0, 4). The line goes through the points (negative 2, 6) and (2, 6).

D: (-∞,∞), R: [4,∞)

The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The vertex is at the point (0, negative 1). The line goes through the points (negative 1, 0) and (1, 0).

The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line segment starts at the point (negative 2, 0). The line goes through the point (0, 2) and ends at the point (2, 0).

D: $\left[-2,2\right],$ R: [0, 2]

The figure has a half-circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The curved line segment starts at the point (negative 3, 3). The line goes through the point (0, 6) and ends at the point (3, 3).

In the following exercises, use the graph of the function to find the indicated values.

This figure has a wavy curved line graphed on the x y-coordinate plane. The x-axis runs from negative 2 times pi to 2 times pi. The y-axis runs from negative 6 to 6. The curved line segment goes through the points (negative 2 times pi, 0), (negative 3 divided by 2 times pi, negative 1), (negative pi, 0), (negative 1 divided by 2 times pi, 1), (0, 0), (1 divided by 2 times pi, negative 1), (pi, 0), (3 divided by 2 times pi, 1), and (2 times pi, 0). The points (negative 3 divided by 2 times pi, negative 1) and (1 divided by 2 times pi, negative 1) are the lowest points on the graph. The points (negative 1 divided by 2 times pi, 1) and (3 divided by 2 times pi, 1) are the highest points on the graph. The pattern extends infinitely to the left and right.

ⓐ Find: $f\left(0\right).$

ⓑ Find: $f\left(\frac{1}{2}\pi \right).$

ⓓ Find the values for x when $f\left(x\right)=0.$

ⓔ Find the x-intercepts.

ⓕ Find the y-intercepts.

ⓖ Find the domain. Write it in interval notation.

ⓗ Find the range. Write it in interval notation.

ⓐ $f\left(0\right)=0$ ⓑ $\left(\text{pi}\text{/}2\right)=-1$

ⓔ $\left(-2\text{pi},0\right),\left(\text{−}\text{pi},0\right),$ $\left(0,0\right),\left(\text{pi},0\right),\left(2\text{pi},0\right)$ $\left(f\right)\left(0,0\right)$

ⓖ $\left[-2\text{pi},2\text{pi}\right]$ ⓗ $\left[-1,1\right]$

ⓐ Find: $f\left(0\right).$

ⓑ Find: $f\left(\pi \right).$

ⓓ Find the values for x when $f\left(x\right)=0.$

ⓔ Find the x-intercepts.

ⓕ Find the y-intercepts.

ⓖ Find the domain. Write it in interval notation.

ⓗ Find the range. Write it in interval notation

The figure has the top half of a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The curved line segment starts at the point (negative 3, 2). The line goes through the point (0, 5) and ends at the point (3, 2). The point (0, 5) is the highest point on the graph. The points (negative 3, 2) and (3, 2) are the lowest points on the graph.

ⓐ Find: $f\left(0\right).$

ⓑ Find: $f\left(-3\right).$

ⓓ Find the values for x when $f\left(x\right)=0.$

ⓔ Find the x-intercepts.

ⓕ Find the y-intercepts.

ⓖ Find the domain. Write it in interval notation.

ⓗ Find the range. Write it in interval notation.

ⓐ $f\left(0\right)=-6$ ⓑ $f\left(-3\right)=3$ ⓒ $f\left(3\right)=3$ ⓓ $f\left(x\right)=0$ for no x ⓔ none ⓕ $y=6$ ⓖ $\left[-3,3\right]$

ⓗ $\left[-3,6\right]$

ⓐ Find: $f\left(0\right).$

ⓑ Find the values for x when $f\left(x\right)=0.$

ⓓ Find the y-intercepts.

ⓔ Find the domain. Write it in interval notation.

ⓕ Find the range. Write it in interval notation

Writing Exercises

Explain in your own words how to find the domain from a graph.

Explain in your own words how to find the range from a graph.

Explain in your own words how to use the vertical line test.

Draw a sketch of the square and cube functions. What are the similarities and differences in the graphs?

Self Check

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

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

Chapter Review Exercises

Graph Linear Equations in Two Variables

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system.

ⓐ $\left(-1,-5\right)$

ⓑ $\left(-3,4\right)$

ⓓ $\left(1,\frac{5}{2}\right)$

This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 5 to 5. The point labeled a is 1 units to the left of the origin and 5 units below the origin and is located in quadrant III. The point labeled b is 3 units to the left of the origin and 4 units above the origin and is located in quadrant II. The point labeled c is 2 units to the right of the origin and 3 units below the origin and is located in quadrant IV. The point labeled d is 1 unit to the right of the origin and 2.5 units above the origin and is located in quadrant I.

ⓐ $\left(-2,0\right)$

ⓑ $\left(0,-4\right)$

ⓓ $\left(3,0\right)$

In the following exercises, determine which ordered pairs are solutions to the given equations.

$5x+y=10;$

ⓐ $\left(5,1\right)$

ⓑ $\left(2,0\right)$

ⓑ, ⓒ

$y=6x-2;$

ⓐ $\left(1,4\right)$

ⓑ $\left(\frac{1}{3},0\right)$

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

$y=4x-3$

This figure shows a straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (negative 1, negative 7), (0, negative 3), (1, negative 1), and (2, 3).

$y=-3x$

$y=\frac{1}{2}x+3$

$y=-\frac{4}{5}x-1$

$x-y=6$

$2x+y=7$

$3x-2y=6$

Graph Vertical and Horizontal lines

In the following exercises, graph each equation.

$y=-2$

$x=3$

This figure shows a vertical straight line graphed on the x y-coordinate plane. The x and y-axes run from negative 8 to 8. The line goes through the points (3, negative 1), (3, 0), and (3, 1).

In the following exercises, graph each pair of equations in the same rectangular coordinate system.

$y=-2x$ and $y=-2$

$y=\frac{4}{3}x$ and $y=\frac{4}{3}$

The figure shows the graphs of a straight horizontal line and a straight slanted line on the same x y-coordinate plane. The x and y axes run from negative 5 to 5. The horizontal line goes through the points (0, 4 divided by 3), (1, 4 divided by 3), and (2, 4 divided by 3). The slanted line goes through the points (0, 0), (1, 4 divided by 3), and (2, 8 divided by 3).

Find x- and y-Intercepts

In the following exercises, find the x– and y-intercepts.

The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (negative 6, negative 2), (negative 4, 0), (negative 2, 2), (0, 4), (2, 6), and (4, 8).

$\left(0,3\right)\left(3,0\right)$

In the following exercises, find the intercepts of each equation.

$x-y=-1$

$x+2y=6$

$\left(6,0\right),\left(0,3\right)$

$2x+3y=12$

$y=\frac{3}{4}x-12$

$\left(16,0\right),\left(0,-12\right)$

$y=3x$

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

$-x+3y=3$

$x-y=4$

$2x-y=5$

The figure shows a straight line graphed on the x y-coordinate plane. The x and y axes run from negative 8 to 8. The line goes through the points (0, negative 5), (1, negative 3), (2, negative 1), and (3, 1).

$2x-4y=8$

$y=4x$

Slope of a Line

Find the Slope of a Line

In the following exercises, find the slope of each line shown.

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (0, 0) and (1, negative 3).

1

$-\frac{1}{2}$

In the following exercises, find the slope of each line.

$y=2$

$x=5$

undefined

$x=-3$

$y=-1$

0

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

$\left(-1,-1\right),\left(0,5\right)$

$\left(3.5\right),\left(4,-1\right)$

$-6$

$\left(-5,-2\right),\left(3,2\right)$

$\left(2,1\right),\left(4,6\right)$

$\frac{5}{2}$

Graph a Line Given a Point and the Slope

In the following exercises, graph each line with the given point and slope.

$\left(2,-2\right);$ $m=\frac{5}{2}$

$\left(-3,4\right);$ $m=-\frac{1}{3}$

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 3, 4) and (0, 3).

x-intercept $-4;$ $m=3$

y-intercept 1; $m=-\frac{3}{4}$

Graph a Line Using Its Slope and Intercept

In the following exercises, identify the slope and y-intercept of each line.

$y=-4x+9$

$y=\frac{5}{3}x-6$

$m=\frac{5}{3};\left(0,-6\right)$

$5x+y=10$

$4x-5y=8$

$m=\frac{4}{5};\left(0,-\frac{8}{5}\right)$

In the following exercises, graph the line of each equation using its slope and y-intercept.

$y=2x+3$

$y=\text{−}x-1$

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (0, negative 1) and (1, negative 2).

$y=-\frac{2}{5}x+3$

$4x-3y=12$

In the following exercises, determine the most convenient method to graph each line.

$x=5$

$y=-3$

horizontal line

$2x+y=5$

$x-y=2$

intercepts

$y=\frac{2}{2}x+2$

$y=\frac{3}{4}x-1$

plotting points

Graph and Interpret Applications of Slope-Intercept

Katherine is a private chef. The equation $C=6.5m+42$ models the relation between her weekly cost, C, in dollars and the number of meals, m, that she serves.

ⓐ Find Katherine’s cost for a week when she serves no meals.

ⓑ Find the cost for a week when she serves 14 meals.

ⓓ Graph the equation.

Marjorie teaches piano. The equation $P=35h-250$ models the relation between her weekly profit, P, in dollars and the number of student lessons, s, that she teaches.

ⓐ Find Marjorie’s profit for a week when she teaches no student lessons.

ⓑ Find the profit for a week when she teaches 20 student lessons.

ⓓ Graph the equation.

ⓐ $\text{−}\text{?}250$

ⓑ ?450

The P-intercept means that when the number of lessons is 0, Marjorie loses ?250.

ⓓ

This figure shows the graph of a straight line on the x y-coordinate plane. The x-axis runs from negative 4 to 28. The y-axis runs from negative 250 to 450. The line goes through the points (0, negative 250) and (20, 450).

Use Slopes to Identify Parallel and Perpendicular Lines

In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.

$4x-3y=-1;y=\frac{4}{3}x-3$

$y=5x-1;10x+2y=0$

neither

$3x-2y=5;2x+3y=6$

$2x-y=8;x-2y=4$

not parallel

Find the Equation of a Line

Find an Equation of the Line Given the Slope and y-Intercept

In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.

slope $\frac{1}{3}$ and $y$ -intercept $\left(0,-6\right)$

slope $-5$ and $y$ -intercept $\left(0,-3\right)$

$y=-5x-3$

slope 0 and $y$ -intercept $\left(0,4\right)$

slope $-2$ and $y$ -intercept $\left(0,0\right)$

$y=-2x$

In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 1), (1, 3), and (2, 5).

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, 5), (1, 2), and (2, negative 1).

$y=-3x+5$

This figure has a graph of a straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 2), (4, 1), and (8, 4).

This figure has a graph of a horizontal straight line on the x y-coordinate plane. The x and y-axes run from negative 10 to 10. The line goes through the points (0, negative 4), (1, negative 4), and (2, negative 4).

$y=-4$

Find an Equation of the Line Given the Slope and a Point

In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.

$m=-\frac{1}{4},$ point $\left(-8,3\right)$

$m=\frac{3}{5},$ point $\left(10,6\right)$

$y=\frac{3}{5}x$

Horizontal line containing $\left(-2,7\right)$

$m=-2,$ point $\left(-1,-3\right)$

$y=-2x-5$

Find an Equation of the Line Given Two Points

In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.

$\left(2,10\right)$ and $\left(-2,-2\right)$

$\left(7,1\right)$ and $\left(5,0\right)$

$y=\frac{1}{2}x-\frac{5}{2}$

$\left(3,8\right)$ and $\left(3,-4\right)$

$\left(5,2\right)$ and $\left(-1,2\right)$

$y=2$

Find an Equation of a Line Parallel to a Given Line

In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.

line $y=-3x+6,$ point $\left(1,-5\right)$

line $2x+5y=-10,$ point $\left(10,4\right)$

$y=-\frac{2}{5}x+8$

line $x=4,$ point $\left(-2,-1\right)$

line $y=-5,$ point $\left(-4,3\right)$

$y=3$

Find an Equation of a Line Perpendicular to a Given Line

In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.

line $y=-\frac{4}{5}x+2,$ point $\left(8,9\right)$

line $2x-3y=9,$ point $\left(-4,0\right)$

$y=-\frac{3}{2}x-6$

line $y=3,$ point $\left(-1,-3\right)$

line $x=-5$ point $\left(2,1\right)$

$y=1$

Graph Linear Inequalities in Two Variables

Verify Solutions to an Inequality in Two Variables

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality $y<x-3\text{:}$

ⓐ $\left(0,1\right)$ ⓑ $\left(-2,-4\right)$ ⓒ $\left(5,2\right)$ ⓓ $\left(3,-1\right)$

ⓔ $\left(-1,-5\right)$

Determine whether each ordered pair is a solution to the inequality $x+y>4\text{:}$

ⓐ $\left(6,1\right)$ ⓑ $\left(-3,6\right)$ ⓒ $\left(3,2\right)$ ⓓ $\left(-5,10\right)$ ⓔ $\left(0,0\right)$

Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

Write the inequality shown by the graph with the boundary line $y=\text{−}x+2.$

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, 2), (1, 1), and (2, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom left half are shaded red to indicate that this is where the solutions of the inequality are.

Write the inequality shown by the graph with the boundary line $y=\frac{2}{3}x-3.$

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 3), (3, negative 1), and (6, 1). The line divides the x y-coordinate plane into two halves. The line and the top left half are shaded red to indicate that this is where the solutions of the inequality are.

$y>\frac{2}{3}x-3$

Write the inequality shown by the shaded region in the graph with the boundary line $x+y=-4.$

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 4), (negative 2, negative 2), and (negative 4, 0). The line divides the x y-coordinate plane into two halves. The line and the top right half are shaded red to indicate that this is where the solutions of the inequality are.

Write the inequality shown by the shaded region in the graph with the boundary line $x-2y=6.$

This figure has the graph of a straight line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A line is drawn through the points (0, negative 3), (2, negative 2), and (6, 0). The line divides the x y-coordinate plane into two halves. The line and the bottom right half are shaded red to indicate that this is where the solutions of the inequality are.

$x-2y\ge 6$

Graph Linear Inequalities in Two Variables

In the following exercises, graph each linear inequality.

Graph the linear inequality $y>\frac{2}{5}x-4.$

Graph the linear inequality $y\le -\frac{1}{4}x+3.$

This figure has the graph of a straight dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 3), (4, 2), and (8, 1). The line divides the x y-coordinate plane into two halves. The bottom left half is shaded red to indicate that this is where the solutions of the inequality are.

Graph the linear inequality $x-y\le 5.$

Graph the linear inequality $3x+2y>10.$

Graph the linear inequality $y\le -3x.$

Graph the linear inequality $y<6.$

This figure has the graph of a straight horizontal dashed line on the x y-coordinate plane. The x and y axes run from negative 10 to 10. A straight dashed line is drawn through the points (0, 6), (1, 6), and (2, 6). The line divides the x y-coordinate plane into two halves. The bottom half is shaded red to indicate that this is where the solutions of the inequality are.

Solve Applications using Linear Inequalities in Two Variables

Shanthie needs to earn at least ?500 a week during her summer break to pay for college. She works two jobs. One as a swimming instructor that pays ?10 an hour and the other as an intern in a law office for ?25 hour. How many hours does Shanthie need to work at each job to earn at least ?500 per week?

ⓐ Let x be the number of hours she works teaching swimming and let y be the number of hours she works as an intern. Write an inequality that would model this situation.

ⓑ Graph the inequality.

Atsushi he needs to exercise enough to burn 600 calories each day. He prefers to either run or bike and burns 20 calories per minute while running and 15 calories a minute while biking.

ⓐ If x is the number of minutes that Atsushi runs and y is the number minutes he bikes, find the inequality that models the situation.

ⓑ Graph the inequality.

ⓐ $20x+15y\ge 600$

ⓑ

The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from 0 to 50. The y-axis runs from 0 to 50. The line goes through the points (0, 40) and (30, 0). The line divides the coordinate plane into two halves. The top right half and the line are colored red to indicate that this is the solution set.

Relations and Functions

Find the Domain and Range of a Relation

In the following exercises, for each relation, ⓐ find the domain of the relation ⓑ find the range of the relation.

$\left\{\left(5,-2\right),\left(5,-4\right),\left(7,-6\right),$

$\left(8,-8\right),\left(9,-10\right)\right\}$

$\left\{\left(-3,7\right),\left(-2,3\right),\left(-1,9\right),$

$\left(0,-3\right),\left(-1,8\right)\right\}$

ⓐ D: {−3, −2, −1, 0}

ⓑ R: {7, 3, 9, −3, 8}

In the following exercise, use the mapping of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

The mapping below shows the average weight of a child according to age.

In the following exercise, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.

The figure shows the graph of some points on the x y-coordinate plane. The x and y-axes run from negative 6 to 6. The points (negative 3, 1), (negative 2, negative 1), (negative 2, negative 3), (0, negative 1), (0, 4), and (4, 3).

ⓐ (4, 3), (−2, −3), (−2, −1), (−3, 1), (0, −1), (0, 4),

ⓑ D: {−3, −2, 0, 4}

Determine if a Relation is a Function

In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function ⓑ find the domain of the relation ⓒ find the range of the relation.

$\left\{\left(9,-5\right),\left(4,-3\right),\left(1,-1\right),$

$\left(0,0\right),\left(1,1\right),\left(4,3\right),\left(9,5\right)\right\}$

$\left\{\left(-3,27\right),\left(-2,8\right),\left(-1,1\right),$

$\left(0,0\right),\left(1,1\right),\left(2,8\right),\left(3,27\right)\right\}$

ⓐ yes ⓑ {−3, −2, −1, 0, 1, 2, 3}

In the following exercises, use the mapping to ⓐ determine whether the relation is a function ⓑ find the domain of the function ⓒ find the range of the function.

This figure shows two table that each have one column. The table on the left has the header “x” and lists the numbers negative 3, negative 2, negative 1, 0, 1, 2, and 3. The table on the right has the header “x to the fourth power” and lists the numbers 0, 1, 16, and 81. There are arrows starting at numbers in the x table and pointing towards numbers in the x to the fourth power table. The first arrow goes from negative 3 to 81. The second arrow goes from negative 2 to 16. The third arrow goes from negative 1 to 1. The fourth arrow goes from 0 to 0. The fifth arrow goes from 1 to 1. The sixth arrow goes from 2 to 16. The seventh arrow goes from 3 to 81.

ⓐ {−3, −2, −1, 0, 1, 2, 3}

ⓑ {−3, −2, −1, 0, 1, 2, 3}

In the following exercises, determine whether each equation is a function.

$2x+y=-3$

$y={x}^{2}$

yes

$y=3x-5$

$y={x}^{3}$

yes

$2x+{y}^{2}=4$

Find the Value of a Function

In the following exercises, evaluate the function:

$f\left(x\right)=3x-4$

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

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

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

In the following exercises, evaluate the function.

$g\left(x\right)=3{x}^{2}-5x;$ $g\left(2\right)$

2

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

$F\left(-1\right)$

$h\left(t\right)=4|t-1|+2;$ $h\left(-3\right)$

18

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

Graphs of Functions

Use the Vertical line Test

In the following exercises, determine whether each graph is the graph of a function.

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 5), (negative 1, 2), (0, 1), (1, 2), and (2, 5). The lowest point on the graph is (0, 1).

yes

The figure has an s-shaped function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curve goes through the points (negative 1, negative 1), (0, 0), and (1, 1).

The figure has a circle graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The circle goes through the points (negative 5, 0), (5, 0), (0, negative 5), and (0, 5).

no

The figure has a parabola opening to the right graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 0), (negative 1, 1), (negative 1, negative 1), (2, 2), and (2, negative 2). The left-most point on the graph is (negative 2, 0).

The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, negative 1), (0, 0), and (1, 1).

yes

The figure has two curved lines graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line on the left goes through the points (negative 3, 0), (negative 4, 2), and (negative 4, negative 2). The curved line on the right goes through the points (3, 0), (4, 2), and (4, negative 2).

The figure has a sideways absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line bends at the point (0, negative 1) and goes to the right. The line goes through the points (1, 0), (1, negative 2), (2, 1), and (2, negative 3).

no

Identify Graphs of Basic Functions

In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.

$f\left(x\right)=5x+1$

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

ⓐ

The figure has a linear function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The line goes through the points (negative 2, 6), (negative 1, 2), and (0, negative 2).

ⓑ D: (-∞,∞), R: (-∞,∞)

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

$f\left(x\right)=-6$

ⓐ

The figure has a constant function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 8 to 4. The line goes through the points (0, negative 6), (1, negative 6), and (2, negative 6).

ⓑ D: (-∞,∞), R: (-∞,∞)

$f\left(x\right)=2x$

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

ⓐ

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 1, 3), (0, 0), and (1, 3). The lowest point on the graph is (0, 0).

ⓑ D: (-∞,∞), R: (-∞,0]

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

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

ⓐ

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 4 to 8. The parabola goes through the points (negative 2, 6), (negative 1, 3), (0, 2), (1, 3), and (2, 6). The lowest point on the graph is (0, 2).

ⓑ D: (-∞,∞), R: (-∞,∞)

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

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

ⓐ

The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 8. The y-axis runs from negative 2 to 10. The half-line starts at the point (negative 2, 0) and goes through the points (negative 1, 1) and (2, 2).

ⓑ D: [ $-2,$ ∞), R: [0,∞)

$f\left(x\right)=\text{−}|x|$

$f\left(x\right)=|x|+1$

ⓐ

The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 1). The line goes through the points (negative 1, 2) and (1, 2).

ⓑ D: (-∞,∞), R: [1,∞)

Read Information from a Graph of a Function

In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation

The figure has a square root function graphed on the x y-coordinate plane. The x-axis runs from 0 to 10. The y-axis runs from 0 to 10. The half-line starts at the point (1, 0) and goes through the points (2, 1) and (5, 2).

The figure has an absolute value function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The vertex is at the point (0, 2). The line goes through the points (negative 1, 3) and (1, 3).

D: (-∞,∞), R: [2,∞)

The figure has a cubic function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 2, negative 4), (0, 0), and (2, 4).

In the following exercises, use the graph of the function to find the indicated values.

ⓐ Find $f\left(0\right).$

ⓑ Find $f\left(\frac{1}{2}\pi \right).$

ⓓ Find the values for x when $f\left(x\right)=0.$

ⓔ Find the $x$ -intercepts.

ⓕ Find the $y$ -intercepts.

ⓖ Find the domain. Write it in interval notation.

ⓗ Find the range. Write it in interval notation.

ⓐ $f\left(x\right)=0$ ⓑ $f\left(\pi \text{/}2\right)=1$

ⓔ $\left(-2\pi ,0\right),$ $\left(\text{−}\pi ,0\right),$ $\left(0,0\right),$ $\left(\pi ,0\right),$ $\left(2\pi ,0\right)$ $\left(f\right)\left(0,0\right)$

ⓖ $\left[-2\pi ,2\pi \right]$ ⓗ $\left[-1,1\right]$

ⓐ Find $f\left(0\right).$

ⓑ Find the values for x when $f\left(x\right)=0.$

ⓓ Find the $y$ -intercepts.

ⓔ Find the domain. Write it in interval notation.

ⓕ Find the range. Write it in interval notation.

Practice Test

Plot each point in a rectangular coordinate system.

ⓐ $\left(2,5\right)$

ⓑ $\left(-1,-3\right)$

ⓓ $\left(-4,\frac{3}{2}\right)$

ⓔ $\left(5,0\right)$

This figure shows points plotted on the x y-coordinate plane. The x and y axes run from negative 10 to 10. The point labeled a is 2 units to the right of the origin and 5 units above the origin and is located in quadrant I. The point labeled b is 1 unit to the left of the origin and 3 units below the origin and is located in quadrant III. The point labeled c is 2 units above the origin and is located on the y-axis. The point labeled d is 4 units to the left of the origin and 1.5 units above the origin and is located in quadrant II. The point labeled e is 5 units to the right of the origin and is located on the x-axis.

Which of the given ordered pairs are solutions to the equation $3x-y=6?$

Find the slope of each line shown.

ⓐ

The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 5, 2) (0, negative 1), and (5, negative 4).

ⓑ

The figure has a straight vertical line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (2, 0) (2, negative 1), and (2, 1).

ⓐ $-\frac{3}{5}$ ⓑ undefined

Find the slope of the line between the points $\left(5,2\right)$ and $\left(-1,-4\right).$

Graph the line with slope $\frac{1}{2}$ containing the point $\left(-3,-4\right).$

The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 3, negative 4) (negative 1, negative 3), and (1, negative 2).

Find the intercepts of $4x+2y=-8$ and graph.

Graph the line for each of the following equations.

$y=\frac{5}{3}x-1$

$y=\text{−}x$

$y=2$

The figure has a straight horizontal line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 1, 2) (0, 2), and (1, 2).

Find the equation of each line. Write the equation in slope-intercept form.

slope $-\frac{3}{4}$ and $y$ -intercept $\left(0,-2\right)$

$m=2,$ point $\left(-3,-1\right)$

$y=2x+5$

containing $\left(10,1\right)$ and $\left(6,-1\right)$

perpendicular to the line $y=\frac{5}{4}x+2,$ containing the point $\left(-10,3\right)$

$y=-\frac{4}{5}x-5$

Write the inequality shown by the graph with the boundary line $y=\text{−}x-3.$

The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 3, 0), (0, negative 3), and (1, negative 4). The line divides the coordinate plane into two halves. The bottom left half and the line are colored red to indicate that this is the solution set.

Graph each linear inequality.

$y>\frac{3}{2}x+5$

The figure has a straight dashed line graphed on the x y-coordinate plane. The x-axis runs from negative 10 to 10. The y-axis runs from negative 10 to 10. The line goes through the points (negative 2, 2), (0, 5), and (2, 8). The line divides the coordinate plane into two halves. The top left half is colored red to indicate that this is the solution set.

$x-y\ge -4$

$y\le -5x$

The figure has a straight line graphed on the x y-coordinate plane. The x-axis runs from negative 8 to 8. The y-axis runs from negative 8 to 8. The line goes through the points (negative 1, 5), (0, 0), and (1, negative 5). The line divides the coordinate plane into two halves. The bottom left half and the line are colored red to indicate that this is the solution set.

Hiro works two part time jobs in order to earn enough money to meet her obligations of at least ?450 a week. Her job at the mall pays ?10 an hour and her administrative assistant job on campus pays ?15 an hour. How many hours does Hiro need to work at each job to earn at least ?450?

ⓐ Let x be the number of hours she works at the mall and let y be the number of hours she works as administrative assistant. Write an inequality that would model this situation.

ⓑ Graph the inequality .

Use the set of ordered pairs to ⓐ determine whether the relation is a function, ⓑ find the domain of the relation, and ⓒ find the range of the relation.

$\left\{\left(-3,27\right),\left(-2,8\right),\left(-1,1\right),\left(0,0\right),$

$\left(1,1\right),\left(2,8\right),\left(3,27\right)\right\}$

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

For $h\left(y\right)=3|y-1|-3,$ evaluate $h\left(-4\right).$

12

Determine whether the graph is the graph of a function. Explain your answer.

The figure has a cube function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 6 to 6. The curved line goes through the points (negative 1, 1), (0, 2), and (1, 3).

In the following exercises, ⓐ graph each function ⓑ state its domain and range.

Write the domain and range in interval notation.

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

ⓐ

The figure has a square function graphed on the x y-coordinate plane. The x-axis runs from negative 6 to 6. The y-axis runs from negative 2 to 10. The parabola goes through the points (negative 2, 5), (negative 1, 2), (0, 1), (1, 2), and (2, 5). The lowest point on the graph is (0, 1).