1. Section 8.3
Graphs of Functions

2. Objectives Find function values graphically
Find the domain and range of a function graphically
Graph nonlinear functions
Translate graphs of functions
Reflect graphs of functions
Find function values and the domain and range of polynomial functions graphically
Use the vertical line test

3. Objective 1: Find Function Values Graphically
From the graph of a function, we can determine function values. In general, the value of ƒ(a) is given by the y-coordinate of a point on the graph of ƒ with x-coordinate a.

4. EXAMPLE 1
Refer to the graph of function f in the figure (a) a. Find ƒ(–3)
b. Find the value of x for which ƒ(x) = –2.
Strategy
In each case, we will use the information provided by the function notation to locate a specific point on the graph and determine its x- and y-coordinates.
Why
Once we locate the specific point, one of its coordinates will equal the value that we are asked to find.

5. EXAMPLE 1
Refer to the graph of function f in the figure (a) a. Find ƒ(–3)
b. Find the value of x for which ƒ(x) = –2.
Solution
a. To find ƒ(–3), we need to find the y-coordinate of the point on the graph of ƒ whose x-coordinate is (–3). If we draw a vertical line through –3 on the x-axis, as shown in figure (b), the line intersects the graph of ƒ at (–3, 5). Therefore, 5 corresponds to –3, and it follows that ƒ(–3) = 5.

6. EXAMPLE 1
Refer to the graph of function f in the figure (a) a. Find ƒ(–3)
b. Find the value of x for which ƒ(x) = –2.
Solution
b. To find the input value x that has an output value ƒ(x) = –2, we draw a horizontal line through –2 on the y-axis, as shown in figure (c) and note that it intersects the graph of ƒ at (4, –2). Since –2 corresponds to 4, it follows that ƒ(x) = –2 if x = 4.

7. Objective 2: Find the Domain and Range of a Function Graphically
We can find the domain and range of a function from its graph. For example, to find the domain of the linear function graphed in figure (a), we project the graph onto the x-axis. Because the graph of the function extends indefinitely to the left and to the right, the projection includes all the real numbers. Therefore, the domain of the function is the set of real numbers.

8. Objective 2: Find the Domain and Range of a Function Graphically
To find the range of the same linear function, we project the graph onto the y-axis, as shown in figure (b). Because the graph of the function extends indefinitely upward and downward, the projection includes all the real numbers. Therefore, the range of the function is the set of real numbers.

9. Objective 3: Graph Nonlinear Functions
We have seen that the graph of a linear function is a line. We will now consider several examples of nonlinear functions whose graphs are not lines. We will begin with ƒ(x) = x2, called the squaring function.
Another important nonlinear function is ƒ(x) = x3, called the cubing function.
A third nonlinear function is ƒ(x) = | x |, called the absolute value function.

10. EXAMPLE 2
Graph ƒ(x) = x2 and find its domain and range.
Strategy
We will graph the function by creating a table of function values and plotting the corresponding ordered pairs.
Why
After drawing a smooth curve though the plotted points, we will have the graph.

11. EXAMPLE 2 Solution Graph ƒ(x) = x2 and find its domain and range.
To graph the function, we select several x-values and find the corresponding values of ƒ(x). For example, if we select –3 for x, we have
Since ƒ (–3) = 9, the ordered pair (–3,9) lies on the graph of ƒ. In a similar manner, we find the corresponding values of ƒ(x) for six other x-values and list the ordered pairs in the table of values. Then we plot the points and draw a smooth curve through them to get the graph, called a parabola.

12. EXAMPLE 2 Solution Graph ƒ(x) = x2 and find its domain and range.
Because the graph extends indefinitely to the left and to the right, the projection of the graph onto the x-axis includes all the real numbers. See figure (a). This means that the domain of the squaring function is the set of real numbers.
Because the graph extends upward indefinitely from the point (0, 0), the projection of the graph on the y-axis includes only positive real numbers and 0. See figure (b). This means that the range of the squaring function is the set of nonnegative real numbers.

13. Objective 4: Translate graphs of Functions
Vertical Translations: If ƒ is a function and k represents a positive number, then
• The graph of y = ƒ(x) + k is identical to the graph of y = ƒ(x) except that it is translated k units upward.
• The graph of y = ƒ(x) – k is identical to the graph of y = ƒ(x) except that it is translated k units downward.
Horizontal Translations: If ƒ is a function and h represents a positive number, then
• The graph of y = ƒ(x – h) is identical to the
graph of y = ƒ(x) except that it is translated
h units to the right.
• The graph of y = ƒ(x + h) is identical to the graph of y = ƒ(x) except that it is translated h units to the left.

14. EXAMPLE 5 Strategy Strategy Why
Graph: g(x) = | x | + 2
Strategy
Strategy
We will graph g(x) = | x | + 2 by translating (shifting) the graph of ƒ(x) = | x | upward 2 units.
Why
The addition of 2 in g(x) = | x | + 2 causes a vertical shift of the graph of the absolute value function 2 units upward.

15. EXAMPLE 5
Graph: g(x) = | x | + 2
Solution
Each point used to graph ƒ(x) = | x |, which is shown in gray, is shifted 2 units upward to obtain the graph of g(x) = | x | + 2, which is shown in red. This is a vertical translation.

16. Objective 5: Reflect graphs of functions
The following figure shows a table of values for ƒ(x) = x2 and for g(x) = –x2. We note that for a given value of x, the corresponding y-value in the tables are opposites. When graphed, we see that the – sign in g(x) = –x2 has the effect of flipping the graph of ƒ(x) = x2 over the x-axis so that the parabola opens downward. We say that the graph of g(x) = –x2 is a reflection of the graph of ƒ(x) = x2 about the x-axis.

17. EXAMPLE 8
Graph g(x) = –x3
Strategy
We will graph g(x) = –x3 by reflecting the graph of ƒ(x) = x3 about the x-axis.
Why
Because of the – sign in g(x) = –x3, the y–coordinate of each point on the graph of function g is the opposite of the y-coordinate of the corresponding point on the graph ƒ(x) = x3.

18. EXAMPLE 8
Graph g(x) = –x3
Solution
To graph g(x) = –x3 , we use the graph of ƒ(x) = x3. First, we reflect the portion of the graph of ƒ(x) = x3 in quadrant I to quadrant IV, as shown. Then we reflect the portion of the graph of ƒ(x) = x3 in quadrant III to quadrant II.

19. Objective 6: Find function values and the domain and range of polynomial functions graphically
The graphs of polynomial functions of degree 1, such as ƒ(x) = 4x – 1 or ƒ(x) = x + 3, are straight lines.
The graphs of polynomial functions of degree 2, such as ƒ(x) = x2 or ƒ(x) = x2 – 2, are parabolas.
The graphs of polynomial functions of degree 3 or higher are often more complicated. Such graphs are always smooth and continuous. That is, they consist of only rounded curves with no sharp corners, and there are no breaks. 19

20. EXAMPLE 9
Refer to the graph of polynomial function f(x) = x3 – 3x2 – 9x + 2 in figure (a).
a. Find f(2).
b. Find any values of x for which f(x) = –25.
c. Find the domain and range of f.
Strategy For parts a and b, we will use the information provided by the function notation to locate a specific point on the graph and determine its x- and y-coordinates. For part c, we will project the graph onto each axis.
Why Once we locate the specific point, one of its coordinates will equal the value that we are asked to find. Projecting the graph onto the x-axis gives the domain and projecting it onto the y-axis gives the range. 20

21. EXAMPLE 9
Refer to the graph of polynomial function f(x) = x3 – 3x2 – 9x + 2 in figure (a).
a. Find f(2).
b. Find any values of x for which f(x) = –25.
c. Find the domain and range of f.
Solution
a. To find f(2), we need to find the y-coordinate of the point on the graph of f whose x-coordinate is 2. If we draw a vertical line downward, from 2 on the x-axis, as shown in figure (b), the line intersects the graph of f at (2, –20). Therefore, –20 corresponds to 2, and it follows that f(2) = –20. 21

22. EXAMPLE 9
Refer to the graph of polynomial function f(x) = x3 – 3x2 – 9x + 2 in figure (a).
a. Find f(2).
b. Find any values of x for which f(x) = –25.
c. Find the domain and range of f.
Solution
b. To find any input value x that has an output value f(x) = –25, we draw a horizontal line through –25 on the y-axis, as shown in figure (c), and note that it intersects the graph of f at (–3, –25) and (3, –25). Therefore, f(–3) = –25 and f(3) = –25. It follows that the values of x for which f(2) = –25 are
–3 and 3. 22

23. EXAMPLE 9
Refer to the graph of polynomial function f(x) = x3 – 3x2 – 9x + 2 in figure (a).
a. Find f(2).
b. Find any values of x for which f(x) = –25.
c. Find the domain and range of f.
Solution
c. To find the domain of f(x) = x3 – 3x2 – 9x + 2, we project its graph onto the x-axis as shown in figure (d). Because the graph extends indefinitely to the left and right, the projection includes all real numbers. Therefore, the domain of the function is the set of real numbers, which can be written in interval notation as (−∞, ∞).
To determine the range of the same polynomial function, we project the graph onto
the y-axis, as shown in figure (e) below. Because the graph of the function extends
indefinitely upward and downward, the projection includes all real numbers. Therefore
the range of the function is the set of real numbers, written (−∞, ∞). 23

24. Objective 7: Use the Vertical Line Test
If a vertical line intersects a graph in more than one point, the graph is not the graph of a function.

25. EXAMPLE 10
Determine whether the graph in figure (a) and figure (c) is the graph of a function.
Strategy
We will check to see whether any vertical lines intersect the graph more than once.
Why
If any vertical line intersects the graph more than once, it is not the graph of a function.

26. EXAMPLE 10
Determine whether the graph in figure (a) and figure (c) is the graph of a function.
Solution
a. Refer to figure (b). The graph shown in red is not the graph of a function because a vertical line intersects the graph more than once The points of intersection of the graph and the vertical line indicate that two values of y (2.5 and –2.5) correspond to the x-value 3.

27. EXAMPLE 10
Determine whether the graph in figure (a) and figure (c) is the graph of a function.
Solution
b. Refer to figure (d). The graph shown in red is the graph of a function, because no vertical line intersects the graph more than once.