1.3 Graphs of Functions

Reading Function Values from a Graph

The Dow-Jones Industrial Average (DJIA) gives the average of the stock prices of 30 major companies. The graph below shows the DJIA as a function of time during the stock market correction of October 1987. The DJIA is thus $f (t)$ , recorded at noon on day $t$ of October.

The values of the input variable, time, are displayed on the horizontal axis, and the values of the output variable, DJIA, are displayed on the vertical axis. There is no formula that gives the DJIA for a particular day; but it is still a function, defined by its graph. The value of $f (t)$ is specified by the vertical coordinate of the point with the given $t$ -coordinate.

Thus, the coordinates of each point on the graph of the function represent a pair of corresponding values of the two variables.

Write an equation that says that the point $open parenthesis 8, 17 close parenthesis$ lies on the graph of $g$ .

_____

$g open parenthesis 8 close parenthesis equals 17$

Which equation says that the point $open parenthesis 8, 17 close parenthesis$ lies on the graph of $g$ ?

$8 + 17 = g$
$g + 8 = 17$
$g open parenthesis 8 close parenthesis equals 17$
$8 = \frac{17}{g}$

The water level in Lake Huron alters unpredictably over time. The graph below gives the average water level, $L (t)$ , in meters in the year $t$ over a 20-year period. (Source: The Canadian Hydrographic Service)

The coordinates of point $H$ on the graph are $open parenthesis 1997, 176.98 close parenthesis$ . What do the coordinates tell you about the function $L$ ?
_____
The average water level in $2004$ was $176.11$ meters. Write this fact in function notation. What can you say about the graph of $L$ ?
_____

$L open parenthesis 1997 close parenthesis equals 176.98$ ; the average water level was $176.98$ meters in $1997$ .
$L open parenthesis 2004 close parenthesis equals 176.11$ . The point $open parenthesis 2004, 176.11 close parenthesis$ lies on the graph of $L$ .

The coordinates of point $H$ on the graph are $open parenthesis 1997, 176.98 close parenthesis$ . What do the coordinates tell you about the function $L$ ?
The average water level in $2004$ was $176.11$ meters. Write this fact in function notation. What can you say about the graph of $L$ ?

$L open parenthesis 1997 close parenthesis equals 176.98$ ; the average water level was $176.98$ meters in $1997$ .
$L open parenthesis 2004 close parenthesis equals 176.11$ . The point $open parenthesis 2004, 176.11 close parenthesis$ lies on the graph of $L$ .

Here is another way of describing how a graph depicts a function.

Example 2

The figure shows the graph of a function $g$ .

Find $g open parenthesis minus 2 close parenthesis$ and $g open parenthesis 5 close parenthesis$ .
For what value(s) of $t$ is $g (t) = - 2$ ?
What is the largest, or maximum, value of $g (t)$ ? For what value of $t$ does the function take on its maximum value?
On what intervals is $g$ increasing?

To find $g open parenthesis minus 2 close parenthesis$ , we look for the point with $t$ -coordinate $minus 2$ . The point $open parenthesis minus 2, 0 close parenthesis$ lies on the graph of $g$ , so $g (- 2) = 0$ . Similarly, the point $open parenthesis 5, 1 close parenthesis$ lies on the graph, so $g open parenthesis 5 close parenthesis equals 1$ .
We look for points on the graph with $y$ -coordinate $minus 2$ . Because the points $(- 5, - 2)$ , $(- 3, - 2)$ , and $open parenthesis 3, minus 2 close parenthesis$ lie on the graph, we know that $g (- 5) = - 2$ , $g (- 3) = - 2$ , and $g (3) = - 2$ . Thus, the $t$ -values we want are $minus 5$ , $minus 3$ , and $3$ .
The highest point on the graph is $open parenthesis 1, 4 close parenthesis$ , so the largest $y$ -value is $4$ . Thus, the maximum value of $g (t)$ is $4$ , and it occurs when $t equals 1$ .
A graph is increasing if the $y$ -values get larger as we read from left to right. The graph of $g$ is increasing for $t$ -values between $minus 4$ and $1$ , and between $3$ and $5$ . Thus, $g$ is increasing on the intervals $open parenthesis minus 4, 1 close parenthesis$ and $open parenthesis 3, 5 close parenthesis$ .

Refer to the graph of the function $g$ shown in Example.

$g open parenthesis 0 close parenthesis equals$ _____
List the value(s) of $t$ for which $g open parenthesis t close parenthesis equals 0$ . Separate different values with commas.
_____
What is the smallest, or minimum, value of $g (t)$ ?
Minimum: _____
For what value of $t$ does the function take on its minimum value?
$t =$ _____
Select all the intervals listed below where $g$ is decreasing.
From $minus 5$ to $minus 4$ ? _____
From $minus 5$ to $minus 2$ ? _____
From 2 to 4? _____
From 1 to 3? _____

$3$
$minus 2, 2, 4$
$minus 3$ ; $t = - 4$
$(- 5, - 4)$ and $open parenthesis 1, 3 close parenthesis$

Refer to the graph of the function $g$ shown in Example.

Find $g open parenthesis 0 close parenthesis$ .
For what value(s) of $t$ is $g open parenthesis t close parenthesis equals 0$ ?
What is the smallest, or minimum, value of $g (t)$ ? For what value of $t$ does the function take on its minimum value?
On what intervals is $g$ is decreasing?

$3$
$minus 2, 2, 4$
$minus 3$ ; $t = - 4$
$(- 5, - 4)$ and $open parenthesis 1, 3 close parenthesis$

Which of the following is true?

_____

The maximum value of $y = f (x)$ may occur at two different $x$ -values.

Which of the following statements is true?

It is not possible for the function $y = f (x)$ to take on the same $y$ -value at two different $x$ -values.
The maximum value of $y = f (x)$ may occur at two different $x$ -values.
The maximum value of the function $y = f (x)$ is the largest $x$ -value that appears on the graph.
$f open parenthesis x close parenthesis equals 0$ at the $y$ -intercept of $f (x)$ .

Technology

Finding Coordinates with a Graphing Utility

We can use the TRACE feature to find the coordinates of points on a graph. For example, graph the equation $y = - 2.6 x - 5.4$ in the window

Xmin $= - 5$		Xmax $equals 4.4$
Ymin $= - 20$		Ymax $equals 15$

Press TRACE, and a “bug” begins flashing on the display. The coordinates of the bug appear at the bottom of the display, as shown in the figure. Use the left and right arrows to move the bug along the graph. You can check that the coordinates of the point $open parenthesis 2, minus 10.6 close parenthesis$ do satisfy the equation $y = - 2.6 x - 5.4$ .

The points identified by the Trace bug depend on the window settings. If we want to find the $y$ -coordinate for a particular $x$ -value, we enter the $x$ -coordinate of the desired point and press ENTER.

Constructing the Graph of a Function

Although some functions are defined by their graphs, we can also construct graphs for functions described by tables or equations. We make these graphs the same way we graph equations in two variables: by plotting points whose coordinates satisfy the equation.

Example 3

Graph the function $f (x) = \sqrt{x + 4}$

We choose several convenient values for $x$ and evaluate the function to find the corresponding $f (x)$ -values. For this function we cannot choose $x$ -values less than $minus 4$ , because the square root of a negative number is not a real number.

$f (- 4) = \sqrt{- 4 + 4} = \sqrt{0} = 0$

$f (- 3) = \sqrt{- 3 + 4} = \sqrt{1} = 1$

$f (0) = \sqrt{0 + 4} = \sqrt{4} = 2$

$f (2) = \sqrt{2 + 4} = \sqrt{6} \approx 2.45$

$f (5) = \sqrt{5 + 4} = \sqrt{9} = 3$

The results are shown in the table.

$x$	$f (x)$
$minus 4$	$0$
$minus 3$	$1$
$0$	$2$
$2$	$the square root of 6$
$5$	$3$

Points on the graph have coordinates $(x, f (x))$ , so the vertical coordinate of each point is given by the value of $f (x)$ . We plot the points and connect them with a smooth curve, as shown in the figure. Notice that no points on the graph have $x$ -coordinates less than $minus 4$ .

How do we find the value of $f open parenthesis 3 close parenthesis$ from a graph of $f$ ?

_____

Find 3 on the $x$ -axis, move vertically to the point, then horizontally to the $y$ -axis.

How do we find the value of $f open parenthesis 3 close parenthesis$ from a graph of $f$ ?

Find 3 on the $x$ -axis, move vertically to the corresponding point on the graph, then horizontally to the $y$ -axis.
Find 3 on the $y$ -axis, move horizontally to the corresponding point on the graph, then vertically to the $x$ -axis.
Substitute 3 for $x$ into the formula for the function.
Substitute 3 for $y$ into the formula for the function.

Let $f (x) = func$

Complete the table of values and sketch a graph of the function.

$x$	$minus 2$	$minus 1$	$- \frac{1}{2}$	$0$	$\frac{1}{2}$	$1$	$2$
$f (x)$	_____	_____	_____	_____	_____	_____	_____

$x$	$minus 2$	$minus 1$	$- \frac{1}{2}$	$0$	$\frac{1}{2}$	$1$	$2$
$f (x)$	$minus 10$	$minus 3$	$\frac{- 17}{8}$	$minus 2$	$\frac{- 15}{8}$	$minus 1$	$6$

The graph is shown below.

~Let $f (x) = x^{3} - 2$

Complete the table of values and sketch a graph of the function.

$x$	$minus 2$	$minus 1$	$- \frac{1}{2}$	$0$	$\frac{1}{2}$	$1$	$2$
$f (x)$

We evaluate the function at each value of $x$ . For example,

$f (- 2) = (- 2)^{3} - 2 = - 8 - 2 = - 10$

You can verify the completed table and the graph with your technology tools.

$x$	$minus 2$	$minus 1$	$- \frac{1}{2}$	$0$	$\frac{1}{2}$	$1$	$2$
$f (x)$	$minus 10$	$minus 3$	$\frac{- 17}{8}$	$minus 2$	$\frac{- 15}{8}$	$minus 1$	$6$

How is graphing a function different from graphing an equation?

_____

How is graphing a function different from graphing an equation?

The Vertical Line Test

In a function, two different outputs cannot be related to the same input. This restriction means that two different ordered pairs cannot have the same first coordinate. What does it mean for the graph of the function?

Consider the graph shown in figure (a) below. Every vertical line intersects the graph in at most one point, so there is only one point on the graph for each $x$ -value. This graph represents a function.

In figure (b), however, the line $x equals 2$ intersects the graph at two points, $open parenthesis 2, 1 close parenthesis$ and $open parenthesis 2, 4 close parenthesis$ . Two different $y$ -values, $1$ and $4$ , are related to the same $x$ -value, $2$ . This graph cannot be the graph of a function.

We summarize these observations as follows.

What does the vertical line test tell us?

_____

The vertical line test tells us if the graph is a function.

What does the vertical line test tell us?

If the graph is a vertical line.
If the graph is increasing.
If the graph is decreasing.
If the graph is a function.

Use the vertical line test to determine which of the graphs below represent functions.

_____

Only (b) is a function.

Use the vertical line test to determine which of the graphs below represent functions.

Only (b) is a function.

Graphical Solution of Equations and Inequalities

The graph of an equation in two variables is just a picture of its solutions. When we read the coordinates of a point on the graph, we are reading a pair of $x$ - and $y$ -values that make the equation true.

For example, the point $open parenthesis 2, 7 close parenthesis$ lies on the graph of $y = 2 x + 3$ shown at right, so we know that the ordered pair $open parenthesis 2, 7 close parenthesis$ is a solution of the equation $y = 2 x + 3$ . You can verify algebraically that $x equals 2$ and $y equals 7$ satisfy the equation:

$Does 7 = 2 (2) + 3 ? Yes$

We can also say that $x equals 2$ is a solution of the one-variable equation $2 x + 3 = 7$ . In fact, we can use the graph of $y = 2 x + 3$ to solve the equation $2 x + 3 = k$ for any value of $k$ . Thus, we can use graphs to find solutions to equations in one variable.

Example 5

Use the graph of $y = 285 - 15 x$ to solve the equation $150 = 285 - 15 x$ .

We begin by locating the point $P$ on the graph for which $y equals 150$ , as shown in the figure.

Next we find the $x$ -coordinate of point $P$ by drawing an imaginary line from $P$ straight down to the $x$ -axis. The $x$ -coordinate of $P$ is $x equals 9$ .

Thus, $P$ is the point $open parenthesis 9, 150 close parenthesis$ , and $x equals 9$ when $y equals 150$ . The solution of the equation $150 = 285 - 15 x$ is $x equals 9$ .

You can verify the solution algebraically by substituting $x equals 9$ into the equation:

Does $150 = 285 - 15 (9)$ ?

$285 - 15 (9) = 285 - 135 = 150. Yes$

If $f open parenthesis 8 close parenthesis equals 2$ , what point lies on the graph of $f$ ?

_____

$open parenthesis 8, 2 close parenthesis$

If $f open parenthesis 8 close parenthesis equals 2$ , what point lies on the graph of $f$ ?

Use the graph of $y = 30 - 8 x$ shown above to solve the equation
$30 - 8 x = 50$

$x =$ _____
Verify your solution algebraically.

The point on the graph where the $y$ -coordinate is 50 is the point $open parenthesis minus 2.5, 50 close parenthesis$ , so $x = - 2.5$ is the solution.
We verify that we have an identity when we substitute $x = - 2.5$ into the equation $30 - 8 x = 50$ .
$\begin{aligned} 30 - 8 (- 2.5) & \overset{?}{=} 50 \\ 30 - (- 20) & = 50 & Yes! \end{aligned}$

For part (b):

We verify that we have an identity when we substitute $x = - 2.5$ into the equation $30 - 8 x = 50$ .

$\begin{aligned} 30 - 8 (- 2.5) & \overset{?}{=} 50 \\ 30 - (- 20) & = 50 & Yes! \end{aligned}$

Use the graph of $y = 30 - 8 x$ shown above to solve the equation
$30 - 8 x = 50$
Verify your solution algebraically.

The point on the line where the $y$ -coordinate is $50$ is the point $open parenthesis minus 2.5, 50 close parenthesis$ , so $x = - 2.5$ is the solution.
We verify that we have an identity when we substitute $x = - 2.5$ into the equation $30 - 8 x = 50$ .
$\begin{aligned} 30 - 8 (- 2.5) & \overset{?}{=} 50 \\ 30 - (- 20) & = 50 & Yes! \end{aligned}$

In a similar fashion, we can solve inequalities with a graph.

Consider again the graph of $y = 2 x + 3$ , shown at right. We saw that $x equals 2$ is the solution of the equation $2 x + 3 = 7$ . When we use $x equals 2$ as the input for the function $f (x) = 2 x + 3$ , the output is $y equals 7$ . Which input values for $x$ produce output values greater than $7$ ?

You can see that $x$ -values greater than $2$ produce $y$ -values greater than $7$ , because points on the graph with $x$ -values greater than $2$ have $y$ -values greater than $7$ . Thus, the solutions of the inequality $2 x plus 3 greater than 7$ are $x greater than 2$ . You can verify this result by solving the inequality algebraically.

Example 6

Use the graph of $y = 285 - 15 x$ to solve the inequality

$285 minus 15 x greater than 150$

We begin by locating the point $P$ on the graph for which $y equals 150$ . Its $x$ -coordinate is $x equals 9$ . Now, because $y = 285 - 15 x$ for points on the graph, the inequality

$285 minus 15 x greater than 150$

is equivalent to $y greater than 150$ .

So we are looking for points on the graph with $y$ -coordinate greater than $150$ . These points are shown in red on the graph. The $x$ -coordinates of these points are the $x$ -values that satisfy the inequality. From the graph, we see that the solutions are $x less than 9$ .

You are using a graph of $f (x) = 48 - 3.5 x$ to solve the inequality $48 - 3.5 x > - 15$ . You find that $f (18) = - 15$ . Your answer is:

_____

$x less than 18$

You are using a graph of $f (x) = 48 - 3.5 x$ to solve the inequality $48 - 3.5 x > - 15$ . You find that $f (18) = - 15$ . What is the solution to the inequality?

Here is the graph from Practice 5 (Practice 5).

Use the graph of $y = 30 - 8 x$ above to solve the inequality
$30 minus 8 x less than or equal to 50$

Answer: _____
Note: Use " $<=$ " for the $\leq$ symbol, and use " $>=$ " for $\geq$ .
Solve the inequality algebraically.

As we found in the previous Checkpoint, the point on $y = 30 - 8 x$ where $y equals 50$ occurs when $x = - 2.5$ . The $y$ -coordinates are smaller as we move to the right on the graph, that is, for the points where $x greater than or equal to minus 2.5$ .
$\begin{aligned} 30 - 8 x & \leq 50 & Subtract 30 from both sides. \\ - 8 x & \leq 20 & Divide by - 8 and reverse inequality. \\ x & \geq - 2.5 \end{aligned}$

Here is the graph from Practice 5 (Practice 5).

Use the graph of
$y = 30 - 8 x$
to solve the inequality
$30 minus 8 x less than or equal to 50$
Solve the inequality algebraically.

As we found in the previous Checkpoint, the point on $y = 30 - 8 x$ where $y equals 50$ occurs when $x = - 2.5$ . The $y$ -coordinates are smaller as we move to the right on the graph, that is, for the points where $x greater than or equal to minus 2.5$ .
$\begin{aligned} 30 - 8 x & \leq 50 & Subtract 30 from both sides. \\ - 8 x & \leq 20 & Divide by - 8 and reverse inequality. \\ x & \geq - 2.5 \end{aligned}$

We can also use this graphical technique to solve nonlinear equations and inequalities.

Example 7

Use a graph of $f (x) = - 2 x^{3} + x^{2} + 16 x$ to solve the equation

$- 2 x^{3} + x^{2} + 16 x = 15$

If we sketch in the horizontal line $y equals 15$ , we can see that there are three points on the graph of $f$ that have $y$ -coordinate $15$ , as shown below. The $x$ -coordinates of these points are the solutions of the equation.

From the graph, we see that the solutions are $x = - 3$ , $x equals 1$ , and approximately $x equals 2.5$ . We can verify each solution algebraically.

For example, if $x = - 3$ , we have

$\begin{aligned} f (- 3) & = - 2 (- 3) 3 + (- 3)^{2} + 16 (- 3) \\ = - 2 (- 27) + 9 - 48 \\ = 54 + 9 - 48 = 15 \end{aligned}$

so $minus 3$ is a solution. Similarly, you can check that $x equals 1$ and $x equals 2.5$ are solutions.

Use the graph of $y = \frac{1}{2} n^{2} + 2 n - 10$ shown above to solve

$\frac{1}{2} n^{2} + 2 n - 10 = 6$

and verify your solutions algebraically.

$n =$ _____

Separate different values with a comma.

$minus 8, 4$

Use the graph of $y = \frac{1}{2} n^{2} + 2 n - 10$ shown above to solve

$\frac{1}{2} n^{2} + 2 n - 10 = 6$

and verify your solutions algebraically.

We look for points on the graph that have $y$ -coordinate $6$ . The $n$ -coordinates of those two points are the solutions, $n = - 8$ and $n equals 4$ . To verify the solutions we evaluate the function at $n = - 8$ and $n equals 4$ to find

$\begin{aligned} \frac{1}{2} (- 8)^{2} + 2 - 8 - 10 & = 32 - 16 - 10 = 6 \\ \frac{1}{2} (4)^{2} + 24 - 10 & = 8 + 8 - 10 = 6 \end{aligned}$

When is a function called decreasing?

_____

A function is called decreasing if its $y$ -values decrease when its $x$ -values increase.

When is a function called decreasing?

If its $x$ -values increase from left to right.
If its $y$ -values decrease when its $x$ -values decrease.
If its $y$ -values decrease when its $x$ -values increase.
If the graph lies below the $x$ -axis.

Technology

Using the Trace Feature

You can use the Trace feature on a graphing utility to approximate solutions to equations. Graph the function $f (x)$ in Example in the window

Xmin $= - 4$		Xmax $equals 4$
Ymin $= - 20$		Ymax $equals 40$

and trace along the curve to the point $open parenthesis 2.4680851, 15.512401 close parenthesis$ . We are close to a solution, because the $y$ -value is close to $15$ . Try entering $x$ -values close to $2.4680851$ , for instance, $x equals 2.4$ and $x equals 2.5$ , to find a better approximation for the solution.

Use the graph above from Practice 7 above to solve the inequality

$\frac{1}{2} n^{2} + 2 n - 10 < 6$

Answer: _____

Note: Use interval notation or inequalities.

$minus 8 less than n less than 4$ , or in interval notation, $open parenthesis minus 8, 4 close parenthesis$

Use the graph above from Practice 6 above to solve the inequality

$\frac{1}{2} n^{2} + 2 n - 10 < 6$

In Practice 6 we found that $\frac{1}{2} n^{2} + 2 n - 10 = 6$ when $n = - 8$ and $n equals 4$ . Looking at the graph, we see that points with $y$ -coordinates less than 6 make up the lower portion of the parabola. They have $x$ -coordinates between $n = - 8$ and $n equals 4$ . Thus, the solutions are $minus 8 less than n less than 4$ , or in interval notation, $open parenthesis minus 8, 4 close parenthesis$

Explain how to use the graph of $y = x + \sqrt{x}$ to solve the equation $x + \sqrt{x} = 56$ .

_____

Explain how to use the graph of $y = x + \sqrt{x}$ to solve the equation $x + \sqrt{x} = 56$ .

Section Summary

Vocabulary

Look up the definitions of new terms in the Glossary.

Coordinates
Maximum
Minimum
Interval
Vertical line test
Inequality
Algebraic solution
Graphical solution

CONCEPTS

The point $(a, b)$ lies on the graph of the function $f$ if and only if $f (a) = b$ .
Each point on the graph of the function $f$ has coordinates $(x, f (x))$ for some value of $x$ .
The vertical line test tells us whether a graph represents a function.
We can use a graph to solve equations and inequalities in one variable.

STUDY QUESTIONS

How can you find the value of $f open parenthesis 3 close parenthesis$ from a graph of $f$ ?
If $f open parenthesis 8 close parenthesis equals 2$ , what point lies on the graph of $f$ ?
Explain how to construct the graph of a function from its equation.
Explain how to use the vertical line test.
How can you solve the equation $x + \sqrt{x} = 56$ using the graph of $y = x + \sqrt{x}$ ?

SKILLS

Practice each skill in the Homework problems listed.

Read function values from a graph: #1–8, 17–20, 33–36
Recognize the graph of a function: #9–10, 31 and 32
Construct a table of values and a graph of a function: #11–16
Solve equations and inequalities graphically: #21–30, 41–50

Homework 1.3

In Problems 1–8, use the graphs to answer the questions about the functions.

Find $h open parenthesis minus 3 close parenthesis$ , $h open parenthesis 1 close parenthesis$ , and $h open parenthesis 3 close parenthesis$ .
For what value(s) of $z$ is $h open parenthesis z close parenthesis equals 3$ ?
Find the intercepts of the graph. List the function values given by the intercepts.
What is the maximum value of $h (z)$ ?
For what value(s) of $z$ does $h$ take on its maximum value?
On what intervals is the function increasing? Decreasing?

$minus 2, 0, 5$
$2$
$h (- 2) = 0, h (1) = 0, h (0) = - 2$
$5$
$3$
Increasing: $open parenthesis minus 3, 0 close parenthesis$ and $open parenthesis 1, 3 close parenthesis$ ; decreasing: $open parenthesis 0, 1 close parenthesis$ and $open parenthesis 3, 5 close parenthesis$

Find $G open parenthesis minus 3 close parenthesis$ , $G open parenthesis minus 1 close parenthesis$ , and $G open parenthesis 2 close parenthesis$ .
For what value(s) of $s$ is $G open parenthesis s close parenthesis equals 3$ ?
Find the intercepts of the graph. List the function values given by the intercepts.
What is the minimum value of $G (s)$ ?
For what value(s) of $s$ does $G$ take on its minimum value?
On what intervals is the function increasing? Decreasing?

Find $R open parenthesis 1 close parenthesis$ and $R open parenthesis 3 close parenthesis$ .
For what value(s) of $p$ is $R open parenthesis p close parenthesis equals 2$ ?
Find the intercepts of the graph. List the function values given by the intercepts.
Find the maximum and minimum values of $R (p)$ .
For what value(s) of $p$ does $R$ take on its maximum and minimum values?
On what intervals is the function increasing? Decreasing?

$minus 1, 2$
$3, minus 1.3$
$R (- 2) = 0$ , $R open parenthesis 2 close parenthesis equals 0$ , $R open parenthesis 4 close parenthesis equals 0$ , $R open parenthesis 0 close parenthesis equals 4$
Max: $4$ ; min: $minus 5$
Max at $p equals 0$ ; min at $p equals 5$
Increasing: $open parenthesis minus 3, 0 close parenthesis$ and $open parenthesis 1, 3 close parenthesis$ ; decreasing: $open parenthesis 0, 1 close parenthesis$ and $open parenthesis 3, 5 close parenthesis$

Find $f open parenthesis minus 1 close parenthesis$ and $f open parenthesis 3 close parenthesis$ .
For what value(s) of $t$ is $f open parenthesis t close parenthesis equals 5$ ?
Find the intercepts of the graph. List the function values given by the intercepts.
Find the maximum and minimum values of $f (t)$ .
For what value(s) of $t$ does $f$ take on its maximum and minimum values?
On what intervals is the function increasing? Decreasing?

Find $S open parenthesis 0 close parenthesis$ , $S (\frac{1}{6})$ , and $S open parenthesis minus 1 close parenthesis$ .
Estimate the value of $S (\frac{1}{3})$ from the graph.
For what value(s) of $x$ is $S (x) = - \frac{1}{2}$ ?
Find the maximum and minimum values of $S (x)$ .
For what value(s) of $x$ does $S$ take on its maximum and minimum values?

$0, \frac{1}{2}, 0$
$0.9$
$\frac{- 5}{6}$ , $\frac{- 1}{6}$ , $\frac{7}{6}$ , $\frac{11}{6}$
Max: $1$ ; min: $minus 1$
Max at $x = - 1.5, 0.5$ ; min at $x = - 0.5, 1.5$

Find $C open parenthesis 0 close parenthesis$ , $C (- \frac{1}{3})$ , and $C open parenthesis 1 close parenthesis$ .
Estimate the value of $C (\frac{1}{6})$ from the graph.
For what value(s) of $x$ is $C (x) = \frac{1}{2}$ ?
Find the maximum and minimum values of $C (x)$ .
For what value(s) of $x$ does $C$ take on its maximum and minimum values?

Find $F open parenthesis minus 3 close parenthesis$ , $F open parenthesis minus 2 close parenthesis$ , and $F open parenthesis 2 close parenthesis$ .
For what value(s) of $s$ is $F (s) = - 1$ ?
Find the maximum and minimum values of $F (s)$ .
For what value(s) of $s$ does $F$ take on its maximum and minimum values?

$2, 2, 1$
$- 6 \leq s < - 4$ or $0 less than or equal to s less than 2$
Max: $2$ ; min: $minus 1$
Max for $- 3 \leq s < - 1$ or $3 less than or equal to s less than 5$ ; min for $- 6 \leq s < - 4$ or $0 less than or equal to s less than 2$

Find $P open parenthesis minus 3 close parenthesis$ , $P open parenthesis minus 2 close parenthesis$ , and $P open parenthesis 1 close parenthesis$ .
For what value(s) of $n$ is $P open parenthesis n close parenthesis equals 0$ ?
Find the maximum and minimum values of $P (n)$ .
For what value(s) of $n$ does $P$ take on its maximum and minimum values?

Which of the graphs in Problems 9 and 10 represent functions?

(a) and (d)

In Problems 11–16,

Make a table of values and sketch a graph of the function by plotting points. (Use the suggested $x$ -values.)
Use a graphing utility to graph the function.

Compare the graphing utility's graph with your sketch.

$g (x) = x^{3} + 4$ ; $x = - 2, - 1, \dots, 2$

$h (x) = 2 + \sqrt{x}$ ; $x equals 0, 1, and so on, 9$

$G (x) = \sqrt{4 - x}$ ; $x = - 5, - 4, \dots, 4$

$F (x) = \sqrt{x - 1}$ ; $x equals 1, 2, and so on, 10$

$v (x) = 1 + 6 x - x^{3}$ ; $x = - 3, - 2, \dots, 3$

$w (x) = x^{3} - 8 x$ ; $x = - 4, - 3, \dots, 4$

The graph shows the speed of sound in the ocean as a function of depth, $S = f (d)$ . The speed of sound is affected both by increasing water pressure and by dropping temperature. (Source: Scientific American)

Evaluate $f open parenthesis 1000 close parenthesis$ and explain its meaning.
Solve $f open parenthesis d close parenthesis equals 1500$ and explain its meaning.
At what depth is the speed of sound the slowest, and what is the speed? Write your answer with function notation.
Describe the behavior of $f (d)$ as $d$ increases.

$f open parenthesis 1000 close parenthesis equals 1495$ : The speed of sound at a depth of $1000$ meters is approximately $1495$ meters per second.
$d equals 570$ or $d equals 1070$ : The speed of sound is $1500$ meters per second at both a depth of $570$ meters and a depth of $1070$ meters.
The slowest speed occurs at a depth of about $810$ meters and the speed is about $1487$ meters per second, so $f open parenthesis 810 close parenthesis equals 1487$ .
$f$ increases from about $1533$ to $1541$ in the first $110$ meters of depth, then drops to about $1487$ at $810$ meters, then rises again, passing $1553$ at a depth of about $1600$ meters.

The graph shows the water level (meters) in Lake Superior as a function of time, $L = f (t)$ . (Source: The Canadian Hydrographic Service)

Evaluate $f open parenthesis 1997 close parenthesis$ and explain its meaning.
Solve $f open parenthesis t close parenthesis equals 183.5$ and explain its meaning.
In which two years did Lake Superior reach its highest levels, and what were those levels? Write your answers with function notation.
Over which two-year period did the water level drop the most?

The graph shows the federal debt as a percentage of the gross domestic product (GDP), as a function of time, $D = f (t)$ . (Source: Office of Management and Budget)

Evaluate $f open parenthesis 1985 close parenthesis$ and explain its meaning.
Solve $f open parenthesis t close parenthesis equals 70$ and explain its meaning.
When did the federal debt reach its highest level since 1960, and what was that level? Write your answer with function notation.
What is the longest time interval over which the federal debt was decreasing?

$f open parenthesis 1985 close parenthesis equals 41$ : The federal debt in $1985$ was about $41 %$ of the gross domestic product.
$t equals 1942$ or $t equals 1955$ : The federal debt was $70 %$ of the gross domestic product in $1942$ and $1955$ .
In about $1997$ , the debt was about $67 %$ of the gross domestic product, so $f open parenthesis 1997 close parenthesis approximately equals 67.3$ .
The percentage basically dropped from 1946 to 1973, but there were small rises around 1950, 1954, 1958, and 1968, so the longest time interval was from 1958 to 1967.

The graph shows the elevation (feet) of the 2005 Los Angeles Marathon course as a function of the distance (miles) into the race, $a = f (d)$ . (Source: Los Angeles Times, March 3, 2005)

Evaluate $f open parenthesis 5 close parenthesis$ and explain its meaning.
Solve $f open parenthesis d close parenthesis equals 200$ and explain its meaning.
Where does the marathon course reach its lowest elevation, and what is that elevation? Write your answer with function notation.
Give three intervals over which the elevation is increasing.

The figure shows a graph of $y = - 2 x + 6$ .

Use the graph to find all values of x for which
1. $y equals 12$
2. $y greater than 12$
3. $y less than 12$
Use the graph to solve
1. $- 2 x + 6 = 12$
2. $- 2 x + 6 > 12$
3. $- 2 x + 6 < 12$
Explain why your answers to parts (a) and (b) are the same.

1. $x = - 3$
2. $x less than minus 3$
3. $x greater than minus 3$
1. $x = - 3$
2. $x less than minus 3$
3. $x greater than minus 3$
On the graph of $y = - 2 x + 6$ , a value of $y$ is the same as a value of $- 2 x + 6$ , so parts (a) and (b) are asking for the same $x$ 's.

The figure shows a graph of $y = \frac{- x}{3} - 6$ .

Use the graph to find all values of x for which
1. $y = - 4$
2. $y greater than minus 4$
3. $y less than minus 4$
Use the graph to solve
1. $\frac{- x}{3} - 6 = - 4$
2. $\frac{- x}{3} - 6 > - 4$
3. $\frac{- x}{3} - 6 < - 4$
Explain why your answers to parts (a) and (b) are the same.

In Problems 23 and 24, use the graph to solve the equation or inequality, and then solve algebraically. (To review solving linear inequalities algebraically, see Algebra Skills Refresher.)

The figure shows the graph of $y = 1.4 x - 0.64$ . Solve the following:

$1.4 x - 0.64 = 0.2$
$- 1.2 = 1.4 x - 0.64$
$1.4 x minus 0.64 greater than 0.2$
$- 1.2 > 1.4 x - 0.64$

$x equals 0.6$
$x = - 0.4$
$x greater than 0.6$
$x less than minus 0.4$

The figure shows the graph of $y = - 2.4 x + 2.32$ . Solve the following:

$1.6 = - 2.4 x + 2.32$
$- 2.4 x + 2.32 = 0.4$
$- 2.4 x + 2.32 \geq 1.6$
$0.4 \geq - 2.4 x + 2.32$

For Problems 25–30, use the graphs to estimate solutions to the equations and inequalities.

The figure shows the graph of $g (x) = \frac{12}{2 + x^{2}}$ .

Solve $\frac{12}{2 + x^{2}} = 4$
Solve $1 \leq \frac{12}{2 + x^{2}} \leq 2$

$x = - 1$ or $x equals 1$
Approximately $- 3 \leq x \leq - 2$ or $2 less than or equal to x less than or equal to 3$

The figure shows the graph of $f (x) = \frac{30 \sqrt{x}}{1 + x}$ .

Solve $\frac{30 \sqrt{x}}{1 + x} = 15$
Solve $\frac{30 \sqrt{x}}{1 + x} < 12$

The figure shows a graph of $B = \frac{1}{3} p^{3} - 3 p + 2$ .

Solve $\frac{1}{3} p^{3} - 3 p + 2 = 6$
Solve $\frac{1}{3} p^{3} - 3 p + 2 = 5$
Solve $\frac{1}{3} p^{3} - 3 p + 2 < 1$
What range of values does $B$ have for $p$ between $minus 2.5$ and $0.5$ ?
For what values of $p$ is $B$ increasing?

$3.5$
$- 2.2, - 1.2, 3.4$
$p less than minus 3.1$ or $0.3 less than p less than 2.8$
$0.5 less than B less than 5.5$
$p less than minus 1.7$ or $p greater than 1.7$

The figure shows a graph of $H = t^{3} - 4 t^{2} - 4 t + 12$ .

Solve $t^{3} - 4 t^{2} - 4 t + 12 = - 4$
Solve $t^{3} - 4 t^{2} - 4 t + 12 = 17$
Solve $t^{3} - 4 t^{2} - 4 t + 12 > 6$
Estimate the horizontal and vertical intercepts of the graph.
For what values of $t$ is $H$ increasing?

The figure shows a graph of $M = g (q)$ .

Find all values of q for which
1. $g open parenthesis q close parenthesis equals 0$
2. $g open parenthesis q close parenthesis equals 16$
3. $g open parenthesis q close parenthesis less than 6$
For what values of $q$ is $g (q)$ increasing?

1. $minus 2, 2$
2. $minus 2.8, 0, 2.8$
3. $- 2.5 < q < - 1.25$ or $1.25 less than q less than 2.5$
$minus 2 less than q less than 0$ or $q greater than 2$

The figure shows a graph of $P = f (t)$ .

Find all values of t for which
1. $f open parenthesis t close parenthesis equals 3$
2. $f open parenthesis t close parenthesis greater than 4.5$
3. $2 less than or equal to f open parenthesis t close parenthesis less than or equal to 4$
For what values of $t$ is $f (t)$ decreasing?

Delbert reads the following values from the graph of a function:
$f (- 3) = 5, f (- 1) = 2, f (1) = 0,$

$f (- 1) = - 4, f (- 3) = - 2$
Can his readings be correct? Explain why or why not.
Francine reads the following values from the graph of a function:
$g (- 2) = 6, g (0) = 0, g (2) = 6,$

$g (4) = 0, g (6) = 6$
Can her readings be correct? Explain why or why not.

He has an error: $f open parenthesis minus 3 close parenthesis$ cannot have both the value $5$ and also the value $minus 2$ , and $f open parenthesis minus 1 close parenthesis$ cannot have both values $2$ and $minus 4$ .
Her readings are possible for a function: each input has only one output.

Sketch the graph of a function that has the following values:
$F (- 2) = 3, F (- 1) = 3, F (0) = 3,$

$F (1) = 3, F (2) = 3$
Sketch the graph of a function that has the following values:
$G (- 2) = 1, G (- 1) = 0, G (0) = - 1,$

$G (1) = 0, G (2) = 1$

For Problems 33–36, graph each function in the friendly window

$\begin{aligned} Xmin & = - 9.4 & Xmax = 9.4 \\ Ymin & = - 10 & Ymax = 10 \end{aligned}$

Then answer the questions about the graph. (See Using a Graphing Calculator for an explanation of friendly windows.)

$g (x) = \sqrt{36 - x^{2}}$

Complete the table. (Round values to tenths.)

$x$	$minus 4$	$minus 2$	$3$	$5$
$g (x)$

Find all points on the graph for which $g open parenthesis x close parenthesis equals 3.6$ .

$x$ $minus 4$ $minus 2$ $3$ $5$
$g (x)$ $4.5$ $5.7$ $5.2$ $3.3$
$minus 4.8, 4.8$

$x$	$minus 4$	$minus 2$	$3$	$5$
$g (x)$	$4.5$	$5.7$	$5.2$	$3.3$

$g (x) = \sqrt{x^{2}} - 6$

Complete the table. (Round values to tenths.)

$x$	$minus 8$	$minus 2$	$3$	$6$
$f (x)$

Find all points on the graph for which $f (x) = - 2$ .

$F (x) = 0.5 x^{3} - 4 x$

Estimate the coordinates of the turning points of the graph, that is, where the graph changes from increasing to decreasing or vice versa.
Write an equation of the form $F (a) = b$ for each turning point.

$(- 1.6, 4.352), (1.6, - 4.352)$
$F (- 1.6) = 4.352$ ; $F (1.6) = - 4.352$

$G (x) = 2 + 4 x - x^{3}$

Estimate the coordinates of the turning points of the graph, that is, where the graph changes from increasing to decreasing or vice versa.
Write an equation of the form $G (a) = b$ for each turning point.

For Problems 37–40, graph the function

first using the standard window.
then using the suggested window. Explain how the window alters the appearance of the graph in each case.

$h (x) = \frac{1}{x^{2} + 10}$

$\begin{aligned} Xmin & = - 5 & Xmax = 5 \\ Ymin & = 0 & Ymax = 0.5 \end{aligned}$

The curve cannot be distinguished from the $x$ -axis in the standard window because the values of $y$ are closer to zero than the resolution of the calculator can display. The second window provides sufficient resolution to see the curve.

$H (x) = \sqrt{1 - x^{2}}$

$\begin{aligned} Xmin & = - 2 & Xmax = 2 \\ Ymin & = - 2 & Ymax = 2 \end{aligned}$

$P (x) = (x - 8) (x + 6) (x - 15)$

$\begin{aligned} Xmin & = - 10 & Xmax = 20 \\ Ymin & = - 250 & Ymax = 750 \end{aligned}$

The curve looks like two vertical lines in the standard window because that window covers too small a region of the plane. The second window allows us to see the turning points of the curve.

$p open parenthesis x close parenthesis equals 200 x cubed$

$\begin{aligned} Xmin & = - 5 & Xmax = 5 \\ Ymin & = - 10, 000 & Ymax = 10, 000 \end{aligned}$

For Problems 41–44, graph the equation with the ZInteger setting. (Press ZOOM $6$ ,then ZOOM $8$ ENTER.) Use the graph to answer each question. Use the equation to verify your answers.

Graph $y = 2 x - 3$

For what value of $x$ is $y equals 5$ ?
For what value of $x$ is $y = - 13$ ?
For what values of $x$ is $y greater than minus 1$ ?
For what values of $x$ is $y less than 25$ ?

$x equals 4$
$x = - 5$
$x greater than 1$
$x less than 14$

Graph $y = 4 - 2 x$

For what value of $x$ is $y equals 6$ ?
For what value of $x$ is $y = - 4$ ?
For what values of $x$ is $y greater than minus 12$ ?
For what values of $x$ is $y less than 18$ ?

Graph $y = 6.5 - 1.8 x$

For what value of $x$ is $y = - 13.3$ ?
For what value of $x$ is $y equals 24.5$ ?
For what values of $x$ is $y less than or equal to 15.5$ ?
For what values of $x$ is $y greater than or equal to minus 7.9$ ?

$x equals 11$
$x = - 10$
$x greater than or equal to minus 5$
$x less than or equal to 8$

Graph $y = 0.2 x + 1.4$

For what value of $x$ is $y = - 5.2$ ?
For what value of $x$ is $y equals 2.8$ ?
For what values of $x$ is $y less than or equal to minus 3.2$ ?
For what values of $x$ is $y greater than or equal to 4.4$ ?

For Problems 45–48, graph the equation with the ZInteger setting. Use the graph to solve each equation or inequality. Check your solutions algebraically.

Graph $y = - 0.4 x + 3.7$

Solve $- 0.4 x + 3.7 = 2.1$
Solve $- 0.4 x + 3.7 > - 5.1$

$x equals 4$
$x less than 22$

Graph $y = 0.4 (x - 1.5)$

Solve $0.4 (x - 1.5) = - 8.6$
Solve $0.4 open parenthesis x minus 1.5 close parenthesis less than 8.6$

Graph $y = \frac{2}{3} x - 24$

Solve $\frac{2}{3} x - 24 = - 10 \frac{2}{3}$
Solve $\frac{2}{3} x - 24 \leq - 19 \frac{1}{3}$

$x equals 20$
$x less than or equal to 7$

Graph $y = \frac{80 - 3 x}{5}$ .

Solve $\frac{80 - 3 x}{5} = 22 \frac{3}{5}$ .
Solve $\frac{80 - 3 x}{5} \leq - 9 \frac{2}{5}$ .

Graph $y = 0.01 x^{3} - 0.1 x^{2} - 2.75 x + 15$ .

Use your graph to solve $0.01 x^{3} - 0.1 x^{2} - 2.75 x + 15 = 0$ .
Press Y= and enter $Y sub 2 equals 10$ . Press GRAPH, and you should see the horizontal line $y equals 10$ superimposed on your previous graph. How many solutions does the equation
$0.01 x^{3} - 0.1 x^{2} - 2.75 x + 15 = 10$
have? Estimate each solution to the nearest whole number.

$minus 15, 5, 20$
$minus 13, 2, 22$

Graph $y = 2.5 x - 0.025 x^{2} - 0.005 x^{3}$ .

Use your graph to solve $2.5 x - 0.025 x^{2} - 0.005 x^{3} = 0$ .
Press Y= and enter $Y_{2} = - 5$ . Press GRAPH, and you should see the horizontal line $y = - 5$ superimposed on your previous graph. How many solutions does the equation
$2.5 x - 0.025 x^{2} - 0.005 x^{3} = - 5$
have? Estimate each solution to the nearest whole number.

Modeling, Functions, and Graphs by Katherine Yoshiwara (yoshiwarabooks.org), GNU Free Documentation License 1.2 or later. Adapted for the XYZ HTML edition with the authors' permission (recorded 2026-07-04). License: GFDL-1.2-or-later.