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STRETCHING, SHRINKING, AND REFLECTING GRAPHS
2.3
Vertical Stretching
●
Vertical Shrinking
●
Reflecting Across an Axis
●
Combining Transformations of Graphs
In the previous section, we saw how adding or subtracting a constant can cause a vertical or horizontal shift. Now we will see how multiplying by a constant alters the
graph of a function.
Vertical Stretching
FOR DISCUSSION
T ECH N OLO GY N O T E
By defining Y1 as directed in parts A,
B, and C, and defining Y2, Y3, and Y4
as shown here, you can minimize your
keystrokes. (These graphs will not appear unless Y1 is defined.)
In each group, we give four related functions. Graph the four functions in the first
group (Group A), and then answer the questions regarding those functions. Repeat the
process for Group B and Group C. Use the window specified for each group.
A
5, 5 by 5, 20
y1 x 2
y2 2x 2
y3 3x 2
y4 4x 2
B
5, 15 by 5, 10
y1 x
y2 2x
y3 3x
y4 4x
C
20, 20 by 10, 10
3
y1 x
3
y2 2
x
3
y3 3x
3
y4 4
x
1. How does the graph of y2 compare to the graph of y1?
2. How does the graph of y3 compare to the graph of y1?
3. How does the graph of y4 compare to the graph of y1?
4. If we choose c 4 , how do you think the graph of y5 c y1 would compare to
the graph of y4? Provide support by choosing such a value of c.
In each group of functions in the preceding activity, we started with a basic function y1 and observed how the graphs of functions of the form y c y1 compared with
the graph of y1 for positive values of c that began at 2 and became progressively larger.
In each case, we obtained a vertical stretch of the graph of the basic function with
which we started. These observations can be generalized to any function.
y
y = c · f (x), c > 1
Vertical Stretching of the Graph of a Function
x
0
If c 1, the graph of y c fx is obtained by vertically stretching the
graph of y fx by a factor of c. In general, the larger the value of c,
the greater the stretch.
y = f(x)
FIGURE 28
In Figure 28, we graphically interpret the statement above.
E X A M P L E 1 Recognizing Vertical Stretches
Figure 29 shows the graphs of four functions. The graph labeled y1 is that of the function defined by fx x. The other three functions, y2 , y3 , and y4 , are defined as follows, but not necessarily in the given order:
2.4x,
3.2x,
and
4.3x.
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Stretching, Shrinking, and Reflecting Graphs
27
Determine the correct equation for each graph.
y2
y4
y3
30
– 10
10
–5
y1 = f (x) = |x|
FIGURE 29
Solution The values of c here are 2.4, 3.2, and 4.3. The vertical heights of the
points with the same x-coordinates of the three graphs will correspond to the magnitudes of these c values. Thus, the graph just above y1 x will be that of y 2.4x,
the “highest” graph will be that of y 4.3x, and the graph of y 3.2x will lie “between” the others. Therefore,
y2 4.3x,
y3 2.4x,
and
y4 3.2x.
If we were to trace to any point on the graph of y1 and then move to the other
graphs one by one, we would see that the y-values of the points would be multiplied
by the appropriate values of c. You may wish to experiment with your calculator in
this way.
Vertical Shrinking
FOR DISCUSSION
This discussion parallels the one given earlier in this section. Follow the same
3 1
1
general directions. (Note: The fractions 4 , 2 , and 4 may be entered as their decimal
equivalents when plotting the graphs.)
T EC H N O LO GY N O T E
You can use a screen such as this to
minimize your keystrokes in parts A, B,
and C. Again, Y1 must be defined in
order to obtain the other graphs.
A
5, 5 by 5, 20
y1 x 2
C
10, 10 by 2, 10
y1 x
y2 34 x 2
B
5, 15 by 2, 5
y1 x
y2 34 x
y3 12 x 2
y3 12 x
y3 12 x
y4 14 x 2
y4 14 x
y4 14 x
y2 34 x
1. How does the graph of y2 compare to the graph of y1?
2. How does the graph of y3 compare to the graph of y1?
3. How does the graph of y4 compare to the graph of y1?
4. If we choose 0 c 14 , how do you think the graph of y5 c y1 would compare
to the graph of y4? Provide support by choosing such a value of c.
In this “For Discussion” activity, we began with a basic function y1 and observed
the graphs of y c y1 , as we chose progressively smaller values of c, with
0 c 1. In each case, the graph of y1 was vertically shrunk (or compressed ). These
observations can also be generalized to any function.
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y
Vertical Shrinking of the Graph of a Function
y = f (x)
x
0
y = c · f (x), 0 < c < 1
FIGURE 30
T ECH N OLO GY N O T E
If 0 c 1, the graph of y c fx is obtained by vertically shrinking
the graph of y fx by a factor of c. In general, the smaller the value of
c, the greater the shrink.
Figure 30 shows a graphical interpretation of vertical shrinking.
E X A M P L E 2 Recognizing Vertical Shrinks
Figure 31 shows the graphs of four functions. The graph labeled y1 is that of the function defined by fx x 3. The other three functions, y2 , y3 , and y4 , are defined as follows, but not necessarily in the given order:
.5x 3, .3x 3, and .1x 3.
Determine the correct equation for each graph.
y1 = f(x) = x 3
y3
y4
y2
3
–3
3
This method of defining Y1 and Y2 
using a list of coefficients in Y2  will
allow you to duplicate Figure 31.
–3
FIGURE 31
Solution The smaller the positive value of c, where 0 c 1, the more compressed toward the x-axis the graph will be. Since we have c .5, .3, and .1, the function rules must be as follows:
y2 .1x 3,
y3 .5x 3,
and
y4 .3x 3.
Reflecting Across an Axis
We have seen how graphs can be transformed by shifting, stretching, and shrinking.
We now examine how graphs can be reflected across an axis.
T ECH N OLO GY N O T E
FOR DISCUSSION
In each pair, we give two related functions. Graph y1 fx and y2 fx in the
standard viewing window, and then answer the questions below for each pair.
A
y1 x 2
y2 x 2
By defining Y1 as directed in parts A, B,
C, and D, and defining Y2 as shown
here, you can minimize your keystrokes
B
y1 x
y2 x
C
y1 x
y2 x
D
y1 x 3
y2 x 3
With respect to the x-axis,
1. how does the graph of y2 compare to the graph of y1?
3
3
x compare with the graph of y x, based on
2. how would the graph of y your answer to Item 1? Confirm your answer by graphing.
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T EC H N O LO GY N O T E
29
Again, in each pair, we give two related functions. Graph y1 fx and
y2 fx in the standard viewing window, and then answer the questions below for
each pair.
E
y1 x
y2 x
By defining Y1 as directed in parts E, F,
and G, and defining Y2 as shown here
(using function notation), you can minimize your keystrokes.
Stretching, Shrinking, and Reflecting Graphs
F
y1 x 3
y2 x 3
G
y1 x 4
3
y2 x 4
3
With respect to the y-axis,
3. how does the graph of y2 compare to the graph of y1?
3
3
x compare with the graph of y x, based on
4. how would the graph of y your answer to Item 3? Confirm your answer by graphing.
The results of the preceding discussion can be formally summarized.
Reflecting the Graph of a Function Across an Axis
For a function defined by y f x,
(a) the graph of y fx is a reflection of the graph of f across
the x-axis.
(b) the graph of y fx is a reflection of the graph of f across
the y-axis.
Figure 32 shows how the reflections just described affect the graph of a function
in general.
y
y
y = f (x)
b
(a, b)
y
(2, 6)
b
(7, 6)
(– a, b)
a
x
0
–a
y = f (x)
x
0
(– 4, 0)
y = f (x)
y = f (– x)
(a, b)
0
x
a
(a, –b)
–b
y = – f (x)
(– 1, – 3)
Reflection across the x-axis
Reflection across the y-axis
(a)
FIGURE 33
(b)
FIGURE 32
y
E X A M P L E 3 Applying Reflections across Axes
Figure 33 shows the graph of a function y fx.
(–1, 3)
(a) Sketch the graph of y fx.
(–4, 0)
x
0
y = –f (x)
(2, – 6) (7, – 6)
FIGURE 34
(b) Sketch the graph of y fx.
Solution
(a) We must reflect the graph across the x-axis. This means that if a point a, b lies
on the graph of y fx, then the point a, b must lie on the graph of
y fx. Using the labeled points, we find the graph of y fx in Figure 34.
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y
(b) Here we must reflect the graph across the y-axis, meaning that if a point a, b lies
on the graph of y fx, then the point a, b must lie on the graph of
y fx. Thus, we obtain the graph of y fx as shown in Figure 35.
(– 7, 6) (– 2, 6)
y = f(–x)
x
0
(4, 0)
To illustrate reflections on calculator-generated graphs, observe Figure 36. Figure 36(a) shows that Y1 has been defined by x 2 6x 12 and Y2 Y1, which
means that the graph of Y2 is a reflection across the x-axis. Figure 36(b) shows the
graphs of Y1 and Y2, confirming this fact. Notice that Y3 Y1x, indicating that the
graph of Y3 is a reflection across the y-axis. This is confirmed by Figure 36(c).
(1, – 3)
FIGURE 35
Y1 = x 2 + 6x + 12
Y1 = x 2 + 6x + 12
10
Y2 = Y1 (– x)
10
–10
10
–10
10
–10
–10
(b)
(c)
Y2 = –Y1
(a)
FIGURE 36
What Went WRONG?
To see how negative values of a affect the graph of y ax 2, a student entered
three functions Y1 , Y2 , and Y3 as in the accompanying screen. The calculator
graphed the first two as shown, but gave a syntax error when it attempted to
graph the third.
10
–10
10
What Went Wrong? What must the student do in order to obtain the desired
graph for y 3x 2?
Answers to What Went Wrong?
The student used a subtraction sign to define Y3 rather than a negative sign. Notice the difference between the
signs in Y1 and Y2 as compared to Y3 . The student must re-enter Y3 using a negative sign.
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31
Combining Transformations of Graphs
y1 = x2
The graphs of y1 x 2 and y2 2x 2 are shown in the same viewing window in
Figure 37. In terms of the types of transformations we have studied, the graph of y2 is
obtained by vertically stretching the graph of y1 by a factor of 2 and then reflecting
across the x-axis. Thus, we have a combination of transformations. As you might expect, we can create an infinite number of functions by vertically stretching or shrinking, shifting upward, downward, left, or right, and reflecting across an axis. The next
example investigates examples of this type of function. In determining the order in
which the transformations are made, use the order of operations.
10
–10
Stretching, Shrinking, and Reflecting Graphs
10
–10
y2 = –2x2
FIGURE 37
E X A M P L E 4 Describing a Combination of Transformations of a Graph
(a) Describe how the graph of y 3x 42 5 can be obtained by transforming
the graph of y x 2. Illustrate with a graphing calculator.
(b) Give the equation of the function that would be obtained by starting with the
graph of y x, shifting 3 units to the left, vertically shrinking the graph by
2
a factor of 3 , reflecting across the x-axis, and shifting the graph 4 units downward, in this order. Illustrate with a graphing calculator.
Analytic Solution
Graphing Calculator Solution
(a) The presence of x 4 in the definition of the
function indicates that the graph of y x 2 must
be shifted 4 units to the right. Since the coefficient of x 42 is 3 (a negative number with
absolute value greater than 1), the graph is
stretched vertically by a factor of 3 and then reflected across the x-axis. Finally, the constant 5
indicates that the graph is shifted upward 5 units.
These ideas are summarized below.
2
y = x2
10
–10
y 3x 42 5



➁ Stretch by a
factor of 3.
➃ Shift 5 units
upward.
(b) Shifting 3 units to the left means that x is transformed to x 3. Vertically shrinking by a factor
2
2
of 3 means multiplying x 3 by 3 , and reflect2
2
ing across the x-axis changes 3 to 3 . Finally,
shifting 4 units downward means subtracting 4.
Putting this all together leads to the equation
y
2
x 3 4.
3
10
–10
➀ Shift 4 units
to the right.
y = –3(x – 4) 2 + 5
FIGURE 38





➂ Reflect across
the x-axis.
(a) Figure 38 supports the discussion in the analytic
solution.
(b) Figure 39 supports the discussion in the analytic
solution.
y = |x |
10
–10
10
–10
y = – 23 | x + 3| – 4
FIGURE 39
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CAUTION
The order in which the transformations are made is important. If they are
made in a different order, a different equation can result. See the diagram that follows.
➀ Stretch by a
factor of 2.



➀ Shift 3 units
to the left.
y 2x 3



y 2x 3
➁ Shift 3 units
upward.
➁ Stretch by a
factor of 2.
E X A M P L E 5 Recognizing a Combination of Transformations
Figure 40 shows two views of the graph of y x and another graph illustrating a
combination of transformations. Find the equation of the transformed graph.
y = |x|
y = |x |
3.1
3.1
–4.7
4.7
–4.7
4.7
–3.1
–3.1
(a)
(b)
FIGURE 40
Solution Figure 40(a) shows that the lowest point on the transformed graph has
coordinates 3, 2, indicating that the graph has been shifted 3 units to the right and
2 units downward. Figure 40(b) shows that a point on the right side of the transformed
graph has coordinates 4, 1, and thus the slope of this ray is
m
2 1 3
3.
34
1
Thus, the stretch factor is 3. This information leads to
y 3x 3 2
as the equation of the transformed graph.
2.3
EXERCISES
Write the equation that results in the desired transformation.
1. The squaring function, stretched by a factor of 2
2. The cubing function, shrunk by a factor of 12
3. The square root function, reflected across the y-axis
4. The cube root function, reflected across the x-axis
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Stretching, Shrinking, and Reflecting Graphs
1
3
5. The absolute value function, stretched by a factor of 3 and
reflected across the x-axis
6. The absolute value function, shrunk by a factor of
reflected across the y-axis
and
7. The cubing function, shrunk by a factor of .25 and reflected across the y-axis
8. The square root function, shrunk by a factor of .2 and reflected across the x-axis
Use the concepts of this chapter to draw a rough sketch of the graphs of y1 , y2 , and y3 . Do not plot points. In each case, y2 and y3 can
be graphed by one or more of these: a vertical and/or horizontal shift of the graph of y1 , a vertical stretch or shrink of the graph of
y1 , or a reflection of the graph of y1 across an axis. After you have made your sketches, check by graphing them in an appropriate
viewing window of your calculator.
9. y1 x, y2 x 3, y3 x 3
10. y1 x 3, y2 x 3 4, y3 x 3 4
11. y1 x, y2 x 3, y3 x 3
12. y1 x, y2 x 3, y3 x 3
13. y1 x, y2 x 6, y3 x 6
14. y1 x, y2 2x, y3 2.5x
3
3
3
15. y1 x, y2 x, y3 2
x
16. y1 x 2, y2 x 22 1, y3 x 22
17. y1 x, y2 2x 1 1, y3 1
x 4
2
18. Concept Check Suppose that the graph of y f x is
symmetric with respect to the y-axis and it is reflected
across the y-axis. How will the new graph compare with
the original one?
3
23. The graph of y 6
x 3 can be obtained from the
3
graph of y x by shifting horizontally
units to the
and stretching vertically by a factor of
.
Fill in each blank with the appropriate response. (Remember
that the vertical stretch or shrink factor is positive.)
3
24. The graph of y .5
x 2 can be obtained from the
3
graph of y x by shifting horizontally
units to the
and shrinking vertically by a factor of
.
19. The graph of y 4x 2 can be obtained from the graph of
y x 2 by vertically stretching by a factor of
and reflecting across the
-axis.
20. The graph of y 6x can be obtained from the graph
of y x by vertically stretching by a factor of
and reflecting across the
-axis.
21. The graph of y 14 x 2 3 can be obtained from
the graph of y x by shifting horizontally
units to the
, vertically shrinking by a factor of
, reflecting across the
-axis, and
shifting vertically
units in the
direction.
22. The graph of y 25 x 6 can be obtained from the
graph of y x by reflecting across the
-axis,
vertically shrinking by a factor of
, reflecting
across the
-axis, and shifting vertically
units in the
direction.
Give the equation of each function whose graph is described.
25. The graph of y x 2 is vertically shrunk by a factor of 12 ,
and the resulting graph is shifted 7 units downward.
26. The graph of y x 3 is vertically stretched by a factor of
3. This graph is then reflected across the x-axis. Finally,
the graph is shifted 8 units upward.
27. The graph of y x is shifted 3 units to the right. This
graph is then vertically stretched by a factor of 4.5. Finally,
the graph is shifted 6 units downward.
3
28. The graph of y x is shifted 2 units to the left. This
graph is then vertically stretched by a factor of 1.5. Finally,
the graph is shifted 8 units upward.
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Shown on the left is the graph of Y1 x 22 1 in the standard viewing window of a graphing calculator. Six other functions,
Y2 through Y7 , are graphed according to the rules shown in the screen on the right.
Y1 = (x – 2)2 + 1
10
–10
10
–10
Match each function with its calculator-generated graph from choices A– F first without using a calculator, by applying the techniques
of this chapter. Then, confirm your answer by graphing the function on your calculator.
29. Y2
30. Y3
A.
31. Y4
32. Y5
B.
10
–10
10
–10
10
–10
–10
D.
34. Y7
C.
10
–10
10
33. Y6
E.
10
–10
10
–10
10
–10
F.
10
–10
10
10
–10
10
–10
–10
In Exercises 35 and 36, the graph of y f x has been transformed to the graph of y gx. No shrinking or stretching is involved.
Give the equation of y gx.
y
35.
36.
y = f(x) = x 2
0
y
x
y = g(x)
(4, 3)
(5, –2)
0
y = g (x)
y = f (x) = x 3
x
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35
In Exercises 37–42, each figure shows the graph of a function y f x. Sketch by hand the graphs of the functions in parts (a), (b),
and (c), and answer the question of part (d).
37. (a) y f x (b) y f x (c) y 2f x
(d) What is f 0?
38. (a) y f x (b) y f x (c) y 3f x
(d) What is f 4?
y
y
y = f (x)
(6, 3)
(–3, 2)
(–2, 1)
(–2, 3)
y = f(x)
(4, 1)
(2, 1) (6, 1)
x
0
x
0
(–4, 0)
39. (a) y f x (b) y f x (c) y f x 1
(d) What are the x-intercepts of y f x 1?
40. (a) y f x (b) y f x (c) y 12 f x
(d) On what interval of the domain is f x 0?
y
y
(2, 32)
y = f(x)
y = f (x)
(3.5, 1.5)
0
0
(–2, 0)
(–1, –3)
x
x
(3, 0)
(0, –2.5)
(– 2, – 32)
41. (a) y f x (b) y f x (c) y .5f x
(d) What symmetry does the graph of y f x exhibit?
42. (a) y f x (b) y f x (c) y 3f x
(d) What symmetry does the graph of y f x exhibit?
y
y
(– 12 , 0) 1
y = f (x)
1
(– 2, 0)
(–1, 0)
(1, 0)
0
x
(– 32 , 0)
(2, 0)
–1
43. Concept Check If r is an x-intercept of the graph of
y f x, what statement can be made about the x-intercept
of the graph of each of the following? (Hint: Make a
sketch.)
(a) y f x
(b) y f x
(c) y f x
( 12 , 0)
x
0
( 32 , 0)
–1
y = f (x)
44. Concept Check If b is the y-intercept of the graph of
y f x, what statement can be made about the y-intercept
of the graph of each of the following? (Hint: Make a
sketch.)
(a) y f x
(b) y f x
(c) y 5f x
(d) y 3f x
Concept Check The sketch shows an example of a function defined by y f x that increases on the interval a, b. Use this graph
as a visual aid, and apply the concepts of reflection introduced in this section to answer each question. (Make your own sketch if you
y
wish.)
y = f(x)
45. Does the graph of y f x increase or decrease on the interval a, b?
46. Does the graph of y f x increase or decrease on the interval b, a?
47. Does the graph of y f x increase or decrease on the interval b, a?
48. If c 0, does the graph of y c f x increase or decrease on the interval a, b?
0
x
a
b
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State the intervals over which each function is (a) increasing, (b) decreasing, and (c) constant.
49. The function graphed in Figure 33
50. The function graphed in Figure 34
51. The function graphed in Figure 35
52. y 2
x 3 4 (See Figure 39.)
3
In Exercises 53–55, each function has a graph with an endpoint (a translation of the point 0, 0). Enter each into your calculator in
an appropriate viewing window, and using your knowledge of the graph of y x , determine the domain and range of the function.
(Hint: Locate the endpoint.)
53. y 10x 20 5
54. y 2x 15 18
55. y .5x 10 5
56. Concept Check Based on your observations in Exercise 53, what are the domain and range of f x ax h k, if
a 0, h 0, and k 0?
3
3
Concept Check Shown here are the graphs of y1 x and y2 5
x. The point whose coordinates are given at the bottom of the
screen lies on the graph of y1. Use this graph, not your calculator, to find the value of y2 for the same value of x shown.
57.
58.
3
y2 = 5√x
3
31
31
– 47
– 47
47
– 31
y2 = 5 √x
47
– 31
3
y1 = √x
3
y1 = √x
Reviewing Basic Concepts (Sections 2.1– 2.3)
1. Suppose that f is defined for all real numbers, and f 3 6. For the given assumptions, find another function value.
(a) The graph of y f x is symmetric with respect to the origin.
(b) The graph of y f x is symmetric with respect to the y-axis.
(c) For all x, f x f x.
(d) For all x, f x f x.
2. Match each equation in Column I with a description of its graph from Column II as it relates to the graph y x 2.
I
(a) y x 72
II
A. a shift of 7 units to the left
(b) y x 2 7
B. a shift of 7 units to the right
(c) y 7x
2
C. a shift of 7 units upward
(d) y x 72
D. a shift of 7 units downward
(e) y x 2 7
E. a vertical stretch by a factor of 7
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3. Match each equation in parts (a)–(h) with the sketch of its graph. The basic graph, y x 2, is shown here.
y
(a) y x 2 2
(b) y x 2 2
y = x2
(c) y x 22
(d) y x 22
(e) y 2x 2
(f) y x 2
(g) y x 22 1
(h) y x 22 1
A.
B.
y
4
0
x
2
C.
y
D.
y
y
4
2
x
0
–2
x
0
2
0
2
0
x
x
2
2
–4
E.
F.
y
G.
y
H.
y
y
5
5
4
4
(–2, 1)
x
0
0
x
–2
2
0
(2, 1)
x
0
4. Match each equation with its calculator-generated graph.
(a) y x 6
(b) y x 6
(d) y x 2 4
(e) y x 2 4
A.
B.
10
–10
–10
10
E.
10
–10
10
–10
10
–10
C.
10
–10
D.
(c) y x 6
10
–10
–10
10
–10
10
–10
10
x
ch02.pgs026-038
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12/3/01
10:39 AM
CHAPTER 2
Page 38
Analysis of Graphs of Functions
5. Each graph is obtained from the graph of f x x or
gx x by applying the transformations discussed in
Sections 2.2 and 2.3. Describe the transformations, and
then give the equation for the graph.
(a)
(b)
y
y
5
1
(c)
1
x
1
x
1
5
(d)
y
y
4
2
2
x
x
4
5
(a) Find a value of c for which gx f x c.
(b) Find a value of c for which gx f x c.
y
y = g(x)
5
3
1
y = f (x)
1 2 3 4 5 6 7
NEXT SECTION
7. Suppose y Fx is changed to y Fx h. How are
the graphs of these equations related? Is the graph of
y Fx h the same as the graph of y Fx h? If
not, how do they differ?
9
8. Suppose the equation y Fx is changed to y c Fx,
for some constant c. What is the effect on the graph of
y Fx? Discuss the effect depending on whether c 0
or c 0, and c 1 or c 1.
9. Complete the table if (a) f is an even function and (b) f is
an odd function.
x
f (x)
3
2
1
1
2
3
4
6
5
10. (Modeling) Carbon Monoxide Levels The 8-hour maximum carbon monoxide levels (in parts per million) for the
United States from 1982 to 1992 can be modeled by the
function defined by
6. Consider the two functions in the figure.
7
9
x
f x .012053x 1.933422 9.07994,
where x 0 corresponds to 1982. (Source: U.S. Environmental Protection Agency, 1992.) Find a function represented by gx that models the same carbon monoxide
levels except that x is the actual year between 1982 and
1992. For example, g1985 f 3 and g1990 f 8.
(Hint: Use a horizontal translation.)