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Modeling, Functions, and Graphs

Katherine Yoshiwara

Section 1.3 Graphs of Functions

Subsection 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 \(f(t)\text{,}\) recorded at noon on day \(t\) of October.
DJIA
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.

Example 1.70.

  1. The coordinates of point \(P\) on the DJIA graph are \((15, 2412)\text{.}\) What do the coordinates tell you about the function \(f\text{?}\)
  2. If the DJIA was 1726 at noon on October 20, what can you say about the graph of \(f\text{?}\)
Solution.
  1. The coordinates of point \(P\) tell us that \(f(15) = 2412\text{,}\) so the DJIA was 2412 at noon on October 15.
  2. We can say that \(f(20) = 1726\text{,}\) so the point \((20, 1726)\) lies on the graph of \(f\text{.}\) This point is labeled \(Q\) in the figure above.
Thus, the coordinates of each point on the graph of the function represent a pair of corresponding values of the two variables.

Graph of a Function.

The point \((a, b)\) lies on the graph of the function \(f\) if and only if \(f(a)=b\text{.}\)

Checkpoint 1.71. QuickCheck 1.

Write an equation that says that the point \((8,17)\) lies on the graph of \(g\text{.}\)
  • \(\displaystyle 8+17=g\)
  • \(\displaystyle g+8=17\)
  • \(\displaystyle g(8)=17\)
  • \(\displaystyle 8=\dfrac{17}{g}\)
Answer.
\(\text{Choice 3}\)
Solution.
\(g(8)=17\)

Checkpoint 1.72. Practice 1.

The water level in Lake Huron alters unpredictably over time. The graph below gives the average water level, \(L(t)\text{,}\) in meters in the year \(t\) over a 20-year period. (Source: The Canadian Hydrographic Service)
graph of Lake Huron water level
  1. The coordinates of point \(H\) on the graph are \((1997, 176.98)\text{.}\) What do the coordinates tell you about the function \(L\text{?}\)
    • \(L(1997) = 176.98\text{;}\) the average water level was 176.98 meters in 1997.
    • \(L(176.98) = 176.98\text{;}\) the average water level was 176.98 meters in 1997.
  2. The average water level in \(2004\) was \(176.11\) meters. Write this fact in function notation. What can you say about the graph of \(L\text{?}\)
    • \(L(176.11) = 2004\text{.}\) The point \((176.11, 2004)\) lies on the graph of \(L\text{.}\)
    • \(L(2004) = 176.11\text{.}\) The point \((2004, 176.11)\) lies on the graph of \(L\text{.}\)
Answer 1.
\(\text{Choice 1}\)
Answer 2.
\(\text{Choice 2}\)
Solution.
  1. \(L(1997) = 176.98\text{;}\) the average water level was \(176.98\) meters in \(1997\text{.}\)
  2. \(L(2004) = 176.11\text{.}\) The point \((2004, 176.11)\) lies on the graph of \(L\text{.}\)
Here is another way of describing how a graph depicts a function.

Functions and Coordinates.

Each point on the graph of the function \(f\) has coordinates \((x, f(x))\) for some value of \(x\text{.}\)

Example 1.73.

The figure shows the graph of a function \(g\text{.}\)
graph of a function
  1. Find \(g(-2)\) and \(g(5)\text{.}\)
  2. For what value(s) of \(t\) is \(g(t) = -2\text{?}\)
  3. What is the largest, or maximum, value of \(g(t)\text{?}\) For what value of \(t\) does the function take on its maximum value?
  4. On what intervals is \(g\) increasing?
Solution.
  1. To find \(g(-2)\text{,}\) we look for the point with \(t\)-coordinate \(-2\text{.}\) The point \((-2, 0)\) lies on the graph of \(g\text{,}\) so \(g(-2) = 0\text{.}\) Similarly, the point \((5, 1)\) lies on the graph, so \(g(5) = 1\text{.}\)
  2. We look for points on the graph with \(y\)-coordinate \(-2\text{.}\) Because the points \((-5, -2)\text{,}\) \((-3, -2)\text{,}\) and \((3, -2)\) lie on the graph, we know that \(g(-5) = -2\text{,}\) \(g(-3) = -2\text{,}\) and \(g(3) = -2\text{.}\) Thus, the \(t\)-values we want are \(-5\text{,}\) \(-3\text{,}\) and \(3\text{.}\)
  3. The highest point on the graph is \((1, 4)\text{,}\) so the largest \(y\)-value is \(4\text{.}\) Thus, the maximum value of \(g(t)\) is \(4\text{,}\) and it occurs when \(t = 1\text{.}\)
  4. 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 \(-4\) and \(1\text{,}\) and between \(3\) and \(5\text{.}\) Thus, \(g\) is increasing on the intervals \((-4, 1)\) and \((3, 5)\text{.}\)

Checkpoint 1.74. Practice 2.

Refer to the graph of the function \(g\) shown in Example 1.73.
graph of a function
  1. \(g(0)=\)
  2. List the value(s) of \(t\) for which \(g(t) = 0 \text{.}\) Separate different values with commas.
  3. What is the smallest, or minimum, value of \(g(t) \text{?}\)
    Minimum:
    For what value of \(t\) does the function take on its minimum value?
    \(t=\)
  4. Select all the intervals listed below where \(g\) is decreasing.
    From \(-5\) to \(-4\text{?}\)
    • Yes
    • No
    From \(-5\) to \(-2\text{?}\)
    • Yes
    • No
    From 2 to 4?
    • Yes
    • No
    From 1 to 3?
    • Yes
    • No
Answer 1.
\(3\)
Answer 2.
\(-2, 2, 4\)
Answer 3.
\(-3\)
Answer 4.
\(-4\)
Answer 5.
\(\text{Yes}\)
Answer 6.
\(\text{No}\)
Answer 7.
\(\text{No}\)
Answer 8.
\(\text{Yes}\)
Solution.
  1. \(\displaystyle 3\)
  2. \(\displaystyle -2, 2, 4\)
  3. \(-3 \text{;}\) \(t = -4\)
  4. \((-5, -4)\) and \((1, 3)\)

Checkpoint 1.75. QuickCheck 2.

Which of the following 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.
  • \(f(x)=0\) at the \(y\)-intercept of \(f(x)\text{.}\)
Answer.
\(\text{Choice 2}\)
Solution.
The maximum value of \(y=f(x)\) may occur at two different \(x\)-values.

Technology 1.76. 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 \(y = -2.6x - 5.4\) in the window
Xmin\(=-5\) Xmax\(=4.4\)
Ymin\(=-20\) Ymax\(=15\)
GC trace
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 \((2, -10.6)\) do satisfy the equation \(y = -2.6x - 5.4\text{.}\)
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.

Subsection 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 1.77.

Graph the function \(f(x) = \sqrt{x + 4}\)
Solution.
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 \(-4\text{,}\) because the square root of a negative number is not a real number.
\begin{equation*} f(\alert{-4}) =\sqrt{\alert{-4} + 4}=\sqrt{0}= 0 \end{equation*}
\begin{equation*} f(\alert{-3}) =\sqrt{\alert{-3} + 4}=\sqrt{1}= 1 \end{equation*}
\begin{equation*} f(\alert{0}) =\sqrt{\alert{0} + 4}=\sqrt{4}=2 \end{equation*}
\begin{equation*} f(\alert{2}) =\sqrt{\alert{2} + 4}=\sqrt{6}\approx 2.45 \end{equation*}
\begin{equation*} f(\alert{5}) =\sqrt{\alert{5} + 4}=\sqrt{9}=3 \end{equation*}
The results are shown in the table.
\(x\) \(f(x)\)
\(-4\) \(0\)
\(-3\) \(1\)
\(0\) \(2\)
\(2\) \(\sqrt{6}\)
\(5\) \(3\)
square root graph
Points on the graph have coordinates \((x, f(x))\text{,}\) so the vertical coordinate of each point is given by the value of \(f(x)\text{.}\) 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 \(-4\text{.}\)

Checkpoint 1.78. QuickCheck 3.

How do we find the value of \(f(3)\) from a graph of \(f\text{?}\)
  • Find 3 on the \(x\)-axis, move vertically to the point, then horizontally to the \(y\)-axis.
  • Find 3 on the \(y\)-axis, move horizontally to the point, then vertically to the \(x\)-axis.
  • Substitue 3 for \(x\) into the formula for the function.
  • Substitue 3 for \(y\) into the formula for the function.
Answer.
\(\text{Choice 1}\)
Solution.
Find 3 on the \(x\)-axis, move vertically to the point, then horizontally to the \(y\)-axis.

Technology 1.79. Using Technology to Graph a Function.

We can also use a graphing utility to obtain a table and graph for the function in Example 1.77. We graph a function just as we graphed an equation. For this function, we enter \(Y_1 = \sqrt{\,}(X+4)\) and press ZOOM \(6\) for the standard window. The graph is shown below.
GC square root graph

Checkpoint 1.80. Practice 3.

Let \(f(x) = {x^{3}-2}\)
Complete the table of values and sketch a graph of the function.
\(x\) \(-2\) \(-1\) \(-\frac{1}{2}\) \(0\) \(\frac{1}{2}\) \(1\) \(2\)
\(f(x)\)
Answer 1.
\(-10\)
Answer 2.
\(-3\)
Answer 3.
\(-{\frac{17}{8}}\)
Answer 4.
\(-2\)
Answer 5.
\(-{\frac{15}{8}}\)
Answer 6.
\(-1\)
Answer 7.
\(6\)
Solution.
\(x\) \(-2\) \(-1\) \(-\frac{1}{2}\) \(0\) \(\frac{1}{2}\) \(1\) \(2\)
\(f(x)\) \(-10\) \(-3\) \(\frac{-17}{8}\) \(-2\) \(\frac{-15}{8}\) \(-1\) \(6\)
The graph is shown below.
cubic

Checkpoint 1.81. Pause and Reflect.

How is graphing a function different from graphing an equation?

Subsection 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.
vertical line test
In figure (b), however, the line \(x = 2\) intersects the graph at two points, \((2, 1)\) and \((2, 4)\text{.}\) Two different \(y\)-values, \(1\) and \(4\text{,}\) are related to the same \(x\)-value, \(2\text{.}\) This graph cannot be the graph of a function.
We summarize these observations as follows.

The Vertical Line Test.

A graph represents a function if and only if every vertical line intersects the graph in at most one point.

Example 1.82.

Use the vertical line test to decide which of the graphs in the figure represent functions.
vertical line test
Solution.
  • Graph (a) represents a function, because it passes the vertical line test.
  • Graph (b) is not the graph of a function, because the vertical line at (for example) \(x = 1\) intersects the graph at two points.
  • For graph (c), notice the break in the curve at \(x = 2\text{:}\) The solid dot at \((2, 1)\) is the only point on the graph with \(x = 2\text{;}\) the open circle at \((2, 3)\) indicates that \((2, 3)\) is not a point on the graph. Thus, graph (c) is a function, with \(f(2) = 1\text{.}\)

Checkpoint 1.83. QuickCheck 4.

What does the vertical line test tell us?
  • A) If the graph is a vertical line.
  • B) If the graph is increasing.
  • C) If the graph is decreasing.
  • D) If the graph is a function.
Answer.
\(\text{D) If ... a function.}\)
Solution.
The vertical line test tells us if the graph is a function.

Checkpoint 1.84. Practice 4.

Use the vertical line test to determine which of the graphs below represent functions.
vertical line test
  • (a)
  • (b)
  • (c)
  • None of the above
Answer.
\(\text{(b)}\)
Solution.
Only (b) is a function.

Subsection 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 \((2, 7)\) lies on the graph of \(y = 2x + 3\) shown at right, so we know that the ordered pair \((2, 7)\) is a solution of the equation \(y = 2x + 3\text{.}\) You can verify algebraically that \(x = \alert{2}\) and \(y = \alert{7}\) satisfy the equation:
\begin{equation*} \text{Does }~\alert{7} = 2 (\alert{2}) + 3\text{?} ~~~~~~~~ \blert{\text{Yes}} \end{equation*}
function graph
We can also say that \(x = 2\) is a solution of the one-variable equation \(2x + 3 = 7\text{.}\) In fact, we can use the graph of \(y = 2x + 3\) to solve the equation \(2x + 3 = k\) for any value of \(k\text{.}\) Thus, we can use graphs to find solutions to equations in one variable.

Example 1.85.

Use the graph of \(y = 285 - 15x\) to solve the equation \(150 = 285 - 15x\text{.}\)
Solution.
We begin by locating the point \(P\) on the graph for which \(y = 150\text{,}\) 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 \(x = 9\text{.}\)
Thus, \(P\) is the point \((9,150)\text{,}\) and \(x = 9\) when \(y = 150\text{.}\) The solution of the equation \(150 = 285 - 15x\) is \(x = 9\text{.}\)
function graph
You can verify the solution algebraically by substituting \(x = \alert{9}\) into the equation:
Does \(150 = 285 - 15(\alert{9})\text{?}\)
\begin{equation*} 285 - 15(\alert{9}) = 285 - 135 = 150. ~~~~~~\blert{\text{Yes}} \end{equation*}

Checkpoint 1.86. QuickCheck 5.

If \(f(8)=2\text{,}\) what point lies on the graph of \(f\text{?}\)
  • \(\displaystyle (8,2)\)
  • \(\displaystyle (2,8)\)
  • Both \((8,2)\) and \((2,8)\)
  • All points between 2 and 8.
Answer.
\(\text{Choice 1}\)
Solution.
\((8,2)\)

Note 1.87.

The relationship between an equation and its graph is an important one. For the previous example, make sure you understand that the following three statements are equivalent:
  • The point \((9, 150)\) lies on the graph of \(y = 285 - 15x\text{.}\)
  • The ordered pair \((9, 150)\) is a solution of the equation \(y = 285 - 15x\text{.}\)
  • \(x = 9\) is a solution of the equation \(150 = 285 - 15x\text{.}\)

Checkpoint 1.88. Practice 5.

function graph
  1. Use the graph of \(y = 30 - 8x\) shown above to solve the equation
    \begin{equation*} 30 - 8x = 50 \end{equation*}
    \(x=\)
  2. Verify your solution algebraically.
Answer.
\(-2.5\)
Solution.
  1. The point on the line where the \(y\)-coordinate is 50 is the point \((-2.5,50)\text{,}\) so \(x=-2.5\) is the solution.
  2. We verify that we have an identity when we substitute \(x=-2.5\) into the equation \(30 - 8x = 50\text{.}\)
    \begin{equation*} \begin{aligned} 30-8(\alert{-2.5}) \amp \stackrel{\text{?}}{=} 50\\ 30 - (-20) \amp = 50 \amp \blert{\text{Yes!}} \end{aligned} \end{equation*}
For part (b):
We verify that we have an identity when we substitute \(x=-2.5\) into the equation \(30 - 8x = 50\text{.}\)
\begin{align*} 30-8(\alert{-2.5}) \amp \stackrel{\text{?}}{=} 50\\ 30 - (-20) \amp = 50 \amp \blert{\text{Yes!}} \end{align*}
In a similar fashion, we can solve inequalities with a graph.
Consider again the graph of \(y = 2x + 3\text{,}\) shown at right. We saw that \(x = 2\) is the solution of the equation \(2x + 3 = 7\text{.}\) When we use \(x = 2\) as the input for the function \(f(x) = 2x + 3\text{,}\) the output is \(y = 7\text{.}\) Which input values for \(x\) produce output values greater than \(7\text{?}\)
You can see that \(x\)-values greater than \(2\) produce \(y\)-values greater than \(7\text{,}\) because points on the graph with \(x\)-values greater than \(2\) have \(y\)-values greater than \(7\text{.}\) Thus, the solutions of the inequality \(2x + 3 \gt 7\) are \(x\gt 2\text{.}\) You can verify this result by solving the inequality algebraically.
function graph

Example 1.89.

Use the graph of \(y = 285 - 15x\) to solve the inequality
\begin{equation*} 285 - 15x \gt 150 \end{equation*}
Solution.
We begin by locating the point \(P\) on the graph for which \(y = 150\text{.}\) Its \(x\)-coordinate is \(x = 9\text{.}\) Now, because \(y = 285 - 15x\) for points on the graph, the inequality
\begin{equation*} 285-15x \gt 150 \end{equation*}
is equivalent to \(y \gt 150\text{.}\)
function graph
So we are looking for points on the graph with \(y\)-coordinate greater than \(150\text{.}\) 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 \(x \lt 9\text{.}\)

Checkpoint 1.90. QuickCheck 6.

You are using a graph of \(f(x)=48-3.5x\) to solve the inequality \(48-3.5x \gt -15\text{.}\) You find that \(f(18)=-15\text{.}\) Your answer is:
  • \(\displaystyle x \gt -15\)
  • \(\displaystyle x \lt -15\)
  • \(\displaystyle x \gt 18\)
  • \(\displaystyle x \lt 18\)
Answer.
\(\text{Choice 4}\)
Solution.
\(x \lt 18\)

Checkpoint 1.91. Practice 6.

Here is the graph from Checkpoint 1.88 (Practice 5).
function graph
  1. Use the graph of \(y = 30 - 8x\) above to solve the inequality
    \begin{equation*} 30 - 8x \le 50 \end{equation*}
    Answer:
    Note: Use “\(\lt=\)” for the \(\le\) symbol, and use “\(\gt=\)” for \(\ge \text{.}\)
  2. Solve the inequality algebraically.
Answer.
\(x\ge -2.5\)
Solution.
  1. As we found in the previous Checkpoint, the point on \(y=30-8x\) where \(y=50\) occurs when \(x=-2.5 \text{.}\) The \(y\)-coordinates are smaller as we move to the right on the graph, that is, for the points where \(x\ge -2.5 \text{.}\)
  2.  
    \begin{equation*} \begin{aligned} 30-8x \amp \le 50 \amp\amp\blert{{\small \text{Subtract 30 from both sides.}}} \\ -8x \amp \le 20 \amp\amp \blert{{\small \text{Divide by } {-8} \text{ and reverse inequality.}}} \\ x \amp \ge -2.5 \end{aligned} \end{equation*}
We can also use this graphical technique to solve nonlinear equations and inequalities.

Example 1.92.

Use a graph of \(f(x) = -2x^3 + x^2 + 16x\) to solve the equation
\begin{equation*} -2x^3 + x^2 + 16x = 15 \end{equation*}
Solution.
If we sketch in the horizontal line \(y = 15\text{,}\) we can see that there are three points on the graph of \(f\) that have \(y\)-coordinate \(15\text{,}\) as shown below. The \(x\)-coordinates of these points are the solutions of the equation.
graph of cubic
From the graph, we see that the solutions are \(x = -3\text{,}\) \(x = 1\text{,}\) and approximately \(x = 2.5\text{.}\) We can verify each solution algebraically.
For example, if \(x = \alert{-3}\text{,}\) we have
\begin{equation*} \begin{aligned}[t] f(\alert{-3})\amp= -2(\alert{-3})3 + (\alert{-3})^2 + 16(\alert{-3})\\ \amp= -2(-27) + 9 - 48\\ \amp =54 + 9 - 48 = 15 \end{aligned} \end{equation*}
so \(-3\) is a solution. Similarly, you can check that \(x = 1\) and \(x = 2.5\) are solutions.

Checkpoint 1.93. Practice 7.

graph of parabola
Use the graph of \(y = \frac{1}{2}n^2 + 2n - 10\) shown above to solve
\begin{equation*} \frac{1}{2}n^2 + 2n - 10 = 6 \end{equation*}
and verify your solutions algebraically.
\(n=\)
Answer.
\(-8, 4\)
Solution.
\(-8, 4\)

Checkpoint 1.94. QuickCheck 7.

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.
Answer.
\(\text{Choice 3}\)
Solution.
A function is called decreasing if its \(y\)-values decrease when its \(x\)-values increase.

Technology 1.95. 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 1.92 in the window
Xmin\(=-4\) Xmax\(=4\)
Ymin\(=-20\) Ymax\(=40\)
and trace along the curve to the point \((2.4680851, 15.512401)\text{.}\) We are close to a solution, because the \(y\)-value is close to \(15\text{.}\) Try entering \(x\)-values close to \(2.4680851\text{,}\) for instance, \(x = 2.4\) and \(x = 2.5\text{,}\) to find a better approximation for the solution.

Example 1.96.

Use the graph in Example 1.92 to solve the inequality
\begin{equation*} -2x^3 + x^2 + 16x \ge 15 \end{equation*}
Solution.
We first locate all points on the graph that have \(y\)-coordinates greater than or equal to \(15\text{.}\) The \(x\)-coordinates of these points are the solutions of the inequality.
The figure below shows the points in red, and their \(x\)-coordinates as intervals on the \(x\)-axis. The solutions are \(x \le -3\) and \(1\le x \le 2.5\text{,}\) or in interval notation, \((-\infty, -3] \cup [1, 2.5]\text{.}\) (See Appendix A.2 to review interval notation.)
graph of cubic

Checkpoint 1.97. Practice 8.

graph of cubic
Use the graph above from Practice 7 above to solve the inequality
\begin{equation*} \begin{gathered} \frac{1}{2}n^2 + 2n - 10 \lt 6 \end{gathered} \end{equation*}
Answer:
Answer.
\(-8 < n < 4\)
Solution.
\(-8 \lt n \lt 4 \text{,}\) or in interval notation, \((-8,4)\)

Checkpoint 1.98. Pause and Reflect.

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

Subsection Section Summary

Subsubsection Vocabulary

Look up the definitions of new terms in the Glossary.
  • Coordinates
  • Maximum
  • Minimum
  • Interval
  • Vertical line test
  • Inequality
  • Algebraic solution
  • Graphical solution

Subsubsection CONCEPTS

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

Subsubsection STUDY QUESTIONS

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

Subsubsection SKILLS

Practice each skill in the Homework problems listed.
  1. Read function values from a graph: #1–8, 17–20, 33–36
  2. Recognize the graph of a function: #9–10, 31 and 32
  3. Construct a table of values and a graph of a function: #11–16
  4. Solve equations and inequalities graphically: #21–30, 41–50

Exercises Homework 1.3

Exercise Group.

In Problems 1–8, use the graphs to answer the questions about the functions.
1.
  1. Find \(h(-3)\text{,}\) \(h(1)\text{,}\) and \(h(3)\text{.}\)
  2. For what value(s) of \(z\) is \(h(z) = 3\text{?}\)
  3. Find the intercepts of the graph. List the function values given by the intercepts.
  4. What is the maximum value of \(h(z)\text{?}\)
  5. For what value(s) of \(z\) does \(h\) take on its maximum value?
  6. On what intervals is the function increasing? Decreasing?
curve
2.
  1. Find \(G(-3)\text{,}\) \(G(-1)\text{,}\) and \(G(2)\text{.}\)
  2. For what value(s) of \(s\) is \(G(s) = 3\text{?}\)
  3. Find the intercepts of the graph. List the function values given by the intercepts.
  4. What is the minimum value of \(G(s)\text{?}\)
  5. For what value(s) of \(s\) does \(G\) take on its minimum value?
  6. On what intervals is the function increasing? Decreasing?
curve
3.
  1. Find \(R(1)\) and \(R(3)\text{.}\)
  2. For what value(s) of \(p\) is \(R(p)=2\text{?}\)
  3. Find the intercepts of the graph. List the function values given by the intercepts.
  4. Find the maximum and minimum values of \(R(p)\text{.}\)
  5. For what value(s) of \(p\) does \(R\) take on its maximum and minimum values?
  6. On what intervals is the function increasing? Decreasing?
curve
4.
  1. Find \(f (-1)\) and \(f (3)\text{.}\)
  2. For what value(s) of \(t\) is \(f(t)=5\text{?}\)
  3. Find the intercepts of the graph. List the function values given by the intercepts.
  4. Find the maximum and minimum values of \(f(t)\text{.}\)
  5. For what value(s) of \(t\) does \(f\) take on its maximum and minimum values?
  6. On what intervals is the function increasing? Decreasing?
curve
5.
  1. Find \(S(0)\text{,}\) \(S\left(\dfrac{1}{6}\right)\text{,}\) and \(S(-1)\text{.}\)
  2. Estimate the value of \(S\left(\dfrac{1}{3}\right)\) from the graph.
  3. For what value(s) of \(x\) is \(S(x) = -\dfrac{1}{2}\text{?}\)
  4. Find the maximum and minimum values of \(S(x)\text{.}\)
  5. For what value(s) of \(x\) does \(S\) take on its maximum and minimum values?
curve
6.
  1. Find \(C(0)\text{,}\) \(C\left(-\dfrac{1}{3}\right)\text{,}\) and \(C(1)\text{.}\)
  2. Estimate the value of \(C\left(\dfrac{1}{6}\right)\) from the graph.
  3. For what value(s) of \(x\) is \(C(x) = \dfrac{1}{2}\text{?}\)
  4. Find the maximum and minimum values of \(C(x)\text{.}\)
  5. For what value(s) of \(x\) does \(C\) take on its maximum and minimum values?
curve
7.
  1. Find \(F(-3)\text{,}\) \(F(-2)\text{,}\) and \(F(2)\text{.}\)
  2. For what value(s) of \(s\) is \(F(s) = -1\text{?}\)
  3. Find the maximum and minimum values of \(F(s)\text{.}\)
  4. For what value(s) of \(s\) does \(F\) take on its maximum and minimum values?
curve
8.
  1. Find \(P(-3)\text{,}\) \(P(-2)\text{,}\) and \(P(1)\text{.}\)
  2. For what value(s) of \(n\) is \(P(n) = 0\text{?}\)
  3. Find the maximum and minimum values of \(P(n)\text{.}\)
  4. For what value(s) of \(n\) does \(P\) take on its maximum and minimum values?
curve

Exercise Group.

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

Exercise Group.

In Problems 11–16,
  1. Make a table of values and sketch a graph of the function by plotting points. (Use the suggested \(x\)-values.)
  2. Use a graphing utility to graph the function.
Compare the graphing utility’s graph with your sketch.
11.
\(g(x) = x^3 + 4\text{;}\) \(\hphantom{00000}x = -2, -1, \ldots , 2\)
12.
\(h(x) = 2 +\sqrt{x}\text{;}\) \(\hphantom{00000}x = 0,1, \ldots , 9\)
13.
\(G(x) =\sqrt{4 - x}\text{;}\) \(\hphantom{00000}x = -5, -4, \ldots , 4\)
14.
\(F(x) = \sqrt{x-1}\text{;}\) \(\hphantom{00000}x = 1,2, \ldots , 10\)
15.
\(v(x) = 1 + 6x - x^3\text{;}\) \(\hphantom{00000}x = -3, -2, \ldots , 3\)
16.
\(w(x) = x^3 - 8x\text{;}\) \(\hphantom{00000}x = -4, -3, \ldots , 4\)

17.

The graph shows the speed of sound in the ocean as a function of depth, \(S = f (d)\text{.}\) The speed of sound is affected both by increasing water pressure and by dropping temperature. (Source: Scientific American)
speed of sound vs depth
  1. Evaluate \(f (1000)\) and explain its meaning.
  2. Solve \(f (d) = 1500\) and explain its meaning.
  3. At what depth is the speed of sound the slowest, and what is the speed? Write your answer with function notation.
  4. Describe the behavior of \(f (d)\) as \(d\) increases.

18.

The graph shows the water level in Lake Superior as a function of time, \(L = f (t)\text{.}\) (Source: The Canadian Hydrographic Service)
water level vs time
  1. Evaluate \(f (1997)\) and explain its meaning.
  2. Solve \(f (t) = 183.5\) and explain its meaning.
  3. In which two years did Lake Superior reach its highest levels, and what were those levels? Write your answers with function notation.
  4. Over which two-year period did the water level drop the most?

19.

The graph shows the federal debt as a percentage of the gross domestic product (GDP), as a function of time, \(D = f (t)\text{.}\) (Source: Office of Management and Budget)
federal debt
  1. Evaluate \(f (1985)\) and explain its meaning.
  2. Solve \(f (t) = 70\) and explain its meaning.
  3. When did the federal debt reach its highest level since 1960, and what was that level? Write your answer with function notation.
  4. What is the longest time interval over which the federal debt was decreasing?

20.

The graph shows the elevation of the Los Angeles Marathon course as a function of the distance into the race, \(a = f (t)\text{.}\) (Source: Los Angeles Times, March 3, 2005)
LA marathon elevation
  1. Evaluate \(f (5)\) and explain its meaning.
  2. Solve \(f (d) = 200\) and explain its meaning.
  3. When does the marathon course reach its lowest elevation, and what is that elevation? Write your answer with function notation.
  4. Give three intervals over which the elevation is increasing.

21.

The figure shows a graph of \(y = -2x + 6\text{.}\)
y=-2x+6
  1. Use the graph to find all values of \(x\) for which
    1. \(\displaystyle y=12\)
    2. \(\displaystyle y\gt 12\)
    3. \(\displaystyle y\lt 12\)
  2. Use the graph to solve
    1. \(\displaystyle -2x + 6=12\)
    2. \(\displaystyle -2x + 6\gt 12\)
    3. \(\displaystyle -2x + 6\lt 12\)
  3. Explain why your answers to parts (a) and (b) are the same.

22.

The figure shows a graph of \(y =\dfrac{-x}{3} - 6\text{.}\)
y=-x/3 - 6
  1. Use the graph to find all values of \(x\) for which
    1. \(\displaystyle y=-4\)
    2. \(\displaystyle y\gt -4\)
    3. \(\displaystyle y\lt -4\)
  2. Use the graph to solve
    1. \(\displaystyle \dfrac{-x}{3} - 6=-4\)
    2. \(\displaystyle \dfrac{-x}{3} - 6\gt -4\)
    3. \(\displaystyle \dfrac{-x}{3} - 6\lt -4\)
  3. Explain why your answers to parts (a) and (b) are the same.

Exercise Group.

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 A.2.)
23.
The figure shows the graph of \(y = 1.4x - 0.64\text{.}\) Solve the following:
y = 1.4x - 0.64
  1. \(\displaystyle 1.4x - 0.64 = 0.2\)
  2. \(\displaystyle -1.2 = 1.4x - 0.64\)
  3. \(\displaystyle 1.4x - 0.64\gt 0.2\)
  4. \(\displaystyle -1.2\gt 1.4x - 0.64\)
24.
The figure shows the graph of \(y = -2.4x + 2.32\text{.}\) Solve the following:
y = -2.4x + 2.32
  1. \(\displaystyle 1.6 = -2.4x + 2.32\)
  2. \(\displaystyle -2.4x + 2.32 = 0.4\)
  3. \(\displaystyle -2.4x + 2.32\ge 1.6\)
  4. \(\displaystyle 0.4\ge -2.4x + 2.32\)

Exercise Group.

For Problems 25–30, use the graphs to estimate solutions to the equations and inequalities.
25.
The figure shows the graph of \(g(x) = \dfrac{12}{2 + x^2}\text{.}\)
bell-like curve
  1. Solve \(\dfrac{12}{2 + x^2} = 4\)
  2. Solve \(1\le \dfrac{12}{2 + x^2} \le 2\)
26.
The figure shows the graph of \(f(x) = \dfrac{30\sqrt{x}}{1 + x}\text{.}\)
curve
  1. Solve \(\dfrac{30\sqrt{x}}{1 + x} = 15\)
  2. Solve \(\dfrac{30\sqrt{x}}{1 + x} \lt 12\)
27.
The figure shows a graph of \(B = \dfrac{1}{3}p^3 - 3p + 2\text{.}\)
cubic curve
  1. Solve \(\dfrac{1}{3}p^3 - 3p + 2 = 6\)
  2. Solve \(\dfrac{1}{3}p^3 - 3p + 2=5\)
  3. Solve \(\dfrac{1}{3}p^3 - 3p + 2\lt 1\)
  4. What range of values does \(B\) have for \(p\) between \(-2.5\) and \(0.5\text{?}\)
  5. For what values of \(p\) is \(B\) increasing?
28.
The figure shows a graph of \(H = t^3 - 4t^2 - 4t + 12\text{.}\)
cubic curve
  1. Solve \(t^3 - 4t^2 - 4t + 12 = -4\)
  2. Solve \(t^3 - 4t^2 - 4t + 12=16\)
  3. Solve \(t^3 - 4t^2 - 4t + 12\gt 6\)
  4. Estimate the horizontal and vertical intercepts of the graph.
  5. For what values of \(t\) is \(H\) increasing?
29.
The figure shows a graph of \(M = g(q)\text{.}\)
mystery curve
  1. Find all values of \(q\) for which
    1. \(\displaystyle g(q) = 0\)
    2. \(\displaystyle g(q) = 16\)
    3. \(\displaystyle g(q)\lt 6\)
  2. For what values of \(q\) is \(g(q)\) increasing?
30.
The figure shows a graph of \(P = f (t)\text{.}\)
mystery curve
  1. Find all values of \(t\) for which
    1. \(\displaystyle f (t) = 3\)
    2. \(\displaystyle f (t)\gt 4.5\)
    3. \(\displaystyle 2\le f (t)\le 4\)
  2. For what values of \(t\) is \(f (t)\) decreasing?

31.

  1. Delbert reads the following values from the graph of a function:
    \begin{equation*} f (-3) = 5, ~f (-1) = 2, ~f (1) = 0, \end{equation*}
    \begin{equation*} f (-1) = -4, ~f (-3) = -2 \end{equation*}
    Can his readings be correct? Explain why or why not.
  2. Francine reads the following values from the graph of a function:
    \begin{equation*} g(-2) = 6, ~g(0) = 0, ~g(2) = 6, \end{equation*}
    \begin{equation*} g(4) = 0, ~g(6) = 6 \end{equation*}
    Can her readings be correct? Explain why or why not.

32.

  1. Sketch the graph of a function that has the following values:
    \begin{equation*} F(-2) = 3, ~F(-1) = 3, ~F(0) = 3, \end{equation*}
    \begin{equation*} F(1) = 3, ~F(2) = 3 \end{equation*}
  2. Sketch the graph of a function that has the following values:
    \begin{equation*} G(-2) = 1, ~G(-1) = 0, ~G(0) = -1, \end{equation*}
    \begin{equation*} G(1) = 0, ~G(2) = 1 \end{equation*}

Exercise Group.

For Problems 33–36, graph each function in the friendly window
\begin{align*} \text{Xmin} \amp = -9.4 \amp\amp \text{Xmax} = 9.4\\ \text{Ymin} \amp = -10 \amp\amp \text{Ymax} = 10 \end{align*}
Then answer the questions about the graph. (See Appendix B for an explanation of friendly windows.)
33.
\(g(x) =\sqrt{36 - x^2}\)
  1. Complete the table. (Round values to tenths.)
    \(x\) \(-4\) \(-2\) \(3\) \(5\)
    \(g(x)\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)
  2. Find all points on the graph for which \(g(x) = 3.6\text{.}\)
34.
\(g(x) =\sqrt{x^2}-6\)
  1. Complete the table. (Round values to tenths.)
    \(x\) \(-8\) \(-2\) \(3\) \(6\)
    \(f(x)\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\) \(\hphantom{000}\)
  2. Find all points on the graph for which \(f(x) = -2\text{.}\)
35.
\(F(x) = 0.5x^3 - 4x\)
  1. Estimate the coordinates of the turning points of the graph, that is, where the graph changes from increasing to decreasing or vice versa.
  2. Write an equation of the form \(F(a) = b\) for each turning point.
36.
\(G(x) = 2 + 4x - x^3\)
  1. Estimate the coordinates of the turning points of the graph, that is, where the graph changes from increasing to decreasing or vice versa.
  2. Write an equation of the form \(G(a) = b\) for each turning point.

Exercise Group.

For Problems 37–40, graph the function
  1. First using the standard window.
  2. Then using the suggested window. Explain how the window alters the appearance of the graph in each case.
37.
\(h(x) = \dfrac{1}{x^2 + 10}\)
\begin{align*} {\text{Xmin}} \amp = -5 \amp\amp {\text{Xmax}} = 5\\ {\text{Ymin}} \amp = 0 \amp\amp {\text{Ymax}} = 0.5 \end{align*}
38.
\(H(x) =\sqrt{1-x^2} \)
\begin{align*} {\text{Xmin}} \amp = -2 \amp\amp {\text{Xmax}} = 2\\ {\text{Ymin}} \amp = -2 \amp\amp {\text{Ymax}} = 2 \end{align*}
39.
\(P(x) = (x - 8)(x + 6)(x - 15)\)
\begin{align*} {\text{Xmin}} \amp = -10 \amp\amp {\text{Xmax}} = 20\\ {\text{Ymin}} \amp = -250 \amp\amp {\text{Ymax}} = 750 \end{align*}
40.
\(p(x) = 200x^3\)
\begin{align*} {\text{Xmin}} \amp = -5 \amp\amp {\text{Xmax}} = 5\\ {\text{Ymin}} \amp = -10,000 \amp\amp {\text{Ymax}} = 10,000 \end{align*}

Exercise Group.

For Problems 41–44, graph the equation with the ZInteger setting. (Press ZOOM \(6\text{,}\)then ZOOM \(8\) ENTER.) Use the graph to answer each question. Use the equation to verify your answers.
41.
Graph \(y = 2x - 3\)
  1. For what value of \(x\) is \(y = 5\text{?}\)
  2. For what value of \(x\) is \(y = -13\text{?}\)
  3. For what values of \(x\) is \(y\gt -1 \text{?}\)
  4. For what values of \(x\) is \(y\lt 25\text{?}\)
42.
Graph \(y = 4 - 2x\)
  1. For what value of \(x\) is \(y = 6\text{?}\)
  2. For what value of \(x\) is \(y = -4\text{?}\)
  3. For what values of \(x\) is \(y\gt -12 \text{?}\)
  4. For what values of \(x\) is \(y\lt 18\text{?}\)
43.
Graph \(y = 6.5 - 1.8x\)
  1. For what value of \(x\) is \(y = -13.3\text{?}\)
  2. For what value of \(x\) is \(y = 24.5\text{?}\)
  3. For what values of \(x\) is \(y\le 15.5 \text{?}\)
  4. For what values of \(x\) is \(y\ge -7.9\text{?}\)
44.
Graph \(y = 0.2x + 1.4\)
  1. For what value of \(x\) is \(y = -5.2\text{?}\)
  2. For what value of \(x\) is \(y = 2.8\text{?}\)
  3. For what values of \(x\) is \(y\le -3.2 \text{?}\)
  4. For what values of \(x\) is \(y\ge 4.4\text{?}\)

Exercise Group.

For Problems 45–48, graph the equation with the ZInteger setting. Use the graph to solve each equation or inequality. Check your solutions algebraically.
45.
Graph \(y = -0.4x + 3.7\)
  1. Solve \(-0.4x + 3.7 = 2.1\)
  2. Solve \(-0.4x + 3.7\gt -5.1\)
46.
Graph \(y = 0.4 (x - 1.5)\)
  1. Solve \(0.4 (x - 1.5) = -8.6\)
  2. Solve \(0.4 (x - 1.5)\lt 8.6\)
47.
Graph \(y = \dfrac{2}{3}x - 24\)
  1. Solve \(\dfrac{2}{3}x - 24 = -10\dfrac{2}{3} \)
  2. Solve \(\dfrac{2}{3}x - 24\le -19\dfrac{1}{3} \)
48.
Graph \(y = \dfrac{80 - 3x}{5}\text{.}\)
  1. Solve \(\dfrac{80 - 3x}{5} = 22\dfrac{3}{5} \text{.}\)
  2. Solve \(\dfrac{80 - 3x}{5}\le -9\dfrac{2}{5} \text{.}\)

49.

Graph \(y = 0.01x^3 - 0.1x^2 - 2.75x + 15\text{.}\)
  1. Use your graph to solve \(0.01x^3 - 0.1x^2 - 2.75x + 15 = 0\text{.}\)
  2. Press Y= and enter \(Y_2 = 10\text{.}\) Press GRAPH, and you should see the horizontal line \(y = 10\) superimposed on your previous graph. How many solutions does the equation
    \begin{equation*} 0.01x^3 - 0.1x^2 - 2.75x + 15 = 10 \end{equation*}
    have? Estimate each solution to the nearest whole number.

50.

Graph \(y = 2.5x - 0.025x^2 - 0.005x^3\text{.}\)
  1. Use your graph to solve \(2.5x - 0.025x^2 - 0.005x^3 = 0\text{.}\)
  2. Press Y= and enter \(Y_2 = -5\text{.}\) Press GRAPH, and you should see the horizontal line \(y = -5\) superimposed on your previous graph. How many solutions does the equation
    \begin{equation*} 2.5x - 0.025x^2 - 0.005x^3 = -5 \end{equation*}
    have? Estimate each solution to the nearest whole number.
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