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.
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.
Example1.70.
The coordinates of point \(P\) on the DJIA graph are \((15, 2412)\text{.}\) What do the coordinates tell you about the function \(f\text{?}\)
If the DJIA was 1726 at noon on October 20, what can you say about the graph of \(f\text{?}\)
The coordinates of point \(P\) tell us that \(f(15) = 2412\text{,}\) so the DJIA was 2412 at noon on October 15.
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 \(f(a)=b\text{.}\)
Checkpoint1.71.QuickCheck 1.
Write an equation that says that the point \((8,17)\) lies on the graph of \(g\text{.}\)
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)
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.
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 \(L\text{.}\)
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 \(g(5) = 1\text{.}\)
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 \(3\text{.}\)
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 \(t = 1\text{.}\)
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 \((3, 5)\text{.}\)
Checkpoint1.74.Practice 2.
Refer to the graph of the function \(g\) shown in Example 1.73.
\(g(0)=\)
List the value(s) of \(t\) for which \(g(t) = 0 \text{.}\) Separate different values with commas.
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?
\(t=\)
Select all the intervals listed below where \(g\) is decreasing.
The maximum value of \(y=f(x)\) may occur at two different \(x\)-values.
Technology1.76.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\(=15\)
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.
SubsectionConstructing 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.
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.
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 \(-4\text{.}\)
Checkpoint1.78.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.
Find 3 on the \(x\)-axis, move vertically to the point, then horizontally to the \(y\)-axis.
Technology1.79.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.
Checkpoint1.80.Practice 3.
Let \(f(x) = {x^{3}-2}\)
Complete the table of values and sketch a graph of the function.
How is graphing a function different from graphing an equation?
SubsectionThe 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 \(x = 2\) 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.
Example1.82.
Use the vertical line test to decide which of the graphs in the figure represent functions.
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 \(f(2) = 1\text{.}\)
SubsectionGraphical 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:
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.
Example1.85.
Use the graph of \(y = 285 - 15x\) to solve the equation \(150 = 285 - 15x\text{.}\)
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 \(x = 9\text{.}\)
You can verify the solution algebraically by substituting \(x = \alert{9}\) into the equation:
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 \(150 = 285 - 15x\text{.}\)
Checkpoint1.88.Practice 5.
Use the graph of \(y = 30 - 8x\) shown above to solve the equation
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.
Example1.89.
Use the graph of \(y = 285 - 15x\) to solve the inequality
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 \(y \gt 150\text{.}\)
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 \(x \lt 9\text{.}\)
Checkpoint1.90.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:
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 \(x\ge -2.5 \text{.}\)
\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.
Example1.92.
Use a graph of \(f(x) = -2x^3 + x^2 + 16x\) to solve the equation
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.
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
A function is called decreasing if its \(y\)-values decrease when its \(x\)-values increase.
Technology1.95.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.
Example1.96.
Use the graph in Example 1.92 to solve the inequality
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.)
Checkpoint1.97.Practice 8.
Use the graph above from Practice 7 above to solve the inequality
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)
Evaluate \(f (1000)\) and explain its meaning.
Solve \(f (d) = 1500\) 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.
18.
The graph shows the water level in Lake Superior as a function of time, \(L = f (t)\text{.}\) (Source: The Canadian Hydrographic Service)
Evaluate \(f (1997)\) and explain its meaning.
Solve \(f (t) = 183.5\) 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?
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)
Evaluate \(f (1985)\) and explain its meaning.
Solve \(f (t) = 70\) 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?
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)
Evaluate \(f (5)\) and explain its meaning.
Solve \(f (d) = 200\) and explain its meaning.
When 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.
21.
The figure shows a graph of \(y = -2x + 6\text{.}\)
Use the graph to find all values of \(x\) for which
\(\displaystyle y=12\)
\(\displaystyle y\gt 12\)
\(\displaystyle y\lt 12\)
Use the graph to solve
\(\displaystyle -2x + 6=12\)
\(\displaystyle -2x + 6\gt 12\)
\(\displaystyle -2x + 6\lt 12\)
Explain why your answers to parts (a) and (b) are the same.
22.
The figure shows a graph of \(y =\dfrac{-x}{3} - 6\text{.}\)
Use the graph to find all values of \(x\) for which
\(\displaystyle y=-4\)
\(\displaystyle y\gt -4\)
\(\displaystyle y\lt -4\)
Use the graph to solve
\(\displaystyle \dfrac{-x}{3} - 6=-4\)
\(\displaystyle \dfrac{-x}{3} - 6\gt -4\)
\(\displaystyle \dfrac{-x}{3} - 6\lt -4\)
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:
\(\displaystyle 1.4x - 0.64 = 0.2\)
\(\displaystyle -1.2 = 1.4x - 0.64\)
\(\displaystyle 1.4x - 0.64\gt 0.2\)
\(\displaystyle -1.2\gt 1.4x - 0.64\)
24.
The figure shows the graph of \(y = -2.4x + 2.32\text{.}\) Solve the following:
\(\displaystyle 1.6 = -2.4x + 2.32\)
\(\displaystyle -2.4x + 2.32 = 0.4\)
\(\displaystyle -2.4x + 2.32\ge 1.6\)
\(\displaystyle 0.4\ge -2.4x + 2.32\)
Exercise Group.
For Problems 25–30, use the graphs to estimate solutions to the equations and inequalities.
25.
The figure shows the graph of \(g(x) = \dfrac{12}{2 + x^2}\text{.}\)
Solve \(\dfrac{12}{2 + x^2} = 4\)
Solve \(1\le \dfrac{12}{2 + x^2} \le 2\)
26.
The figure shows the graph of \(f(x) = \dfrac{30\sqrt{x}}{1 + x}\text{.}\)
Solve \(\dfrac{30\sqrt{x}}{1 + x} = 15\)
Solve \(\dfrac{30\sqrt{x}}{1 + x} \lt 12\)
27.
The figure shows a graph of \(B = \dfrac{1}{3}p^3 - 3p + 2\text{.}\)
Solve \(\dfrac{1}{3}p^3 - 3p + 2 = 6\)
Solve \(\dfrac{1}{3}p^3 - 3p + 2=5\)
Solve \(\dfrac{1}{3}p^3 - 3p + 2\lt 1\)
What range of values does \(B\) have for \(p\) between \(-2.5\) and \(0.5\text{?}\)
For what values of \(p\) is \(B\) increasing?
28.
The figure shows a graph of \(H = t^3 - 4t^2 - 4t + 12\text{.}\)
Solve \(t^3 - 4t^2 - 4t + 12 = -4\)
Solve \(t^3 - 4t^2 - 4t + 12=16\)
Solve \(t^3 - 4t^2 - 4t + 12\gt 6\)
Estimate the horizontal and vertical intercepts of the graph.
For what values of \(t\) is \(H\) increasing?
29.
The figure shows a graph of \(M = g(q)\text{.}\)
Find all values of \(q\) for which
\(\displaystyle g(q) = 0\)
\(\displaystyle g(q) = 16\)
\(\displaystyle g(q)\lt 6\)
For what values of \(q\) is \(g(q)\) increasing?
30.
The figure shows a graph of \(P = f (t)\text{.}\)
Find all values of \(t\) for which
\(\displaystyle f (t) = 3\)
\(\displaystyle f (t)\gt 4.5\)
\(\displaystyle 2\le f (t)\le 4\)
For what values of \(t\) is \(f (t)\) decreasing?
31.
Delbert reads the following values from the graph of a function:
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 \(y = 2x - 3\)
For what value of \(x\) is \(y = 5\text{?}\)
For what value of \(x\) is \(y = -13\text{?}\)
For what values of \(x\) is \(y\gt -1 \text{?}\)
For what values of \(x\) is \(y\lt 25\text{?}\)
42.
Graph \(y = 4 - 2x\)
For what value of \(x\) is \(y = 6\text{?}\)
For what value of \(x\) is \(y = -4\text{?}\)
For what values of \(x\) is \(y\gt -12 \text{?}\)
For what values of \(x\) is \(y\lt 18\text{?}\)
43.
Graph \(y = 6.5 - 1.8x\)
For what value of \(x\) is \(y = -13.3\text{?}\)
For what value of \(x\) is \(y = 24.5\text{?}\)
For what values of \(x\) is \(y\le 15.5 \text{?}\)
For what values of \(x\) is \(y\ge -7.9\text{?}\)
44.
Graph \(y = 0.2x + 1.4\)
For what value of \(x\) is \(y = -5.2\text{?}\)
For what value of \(x\) is \(y = 2.8\text{?}\)
For what values of \(x\) is \(y\le -3.2 \text{?}\)
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.
Use your graph to solve \(0.01x^3 - 0.1x^2 - 2.75x + 15 = 0\text{.}\)
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
have? Estimate each solution to the nearest whole number.
50.
Graph \(y = 2.5x - 0.025x^2 - 0.005x^3\text{.}\)
Use your graph to solve \(2.5x - 0.025x^2 - 0.005x^3 = 0\text{.}\)
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
