7 Working with Functions: Getting Information from the Graph
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SECTION 1.7
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Working with Functions: Getting Information from the Graph
75
e x a m p l e 1 Verbal Description from a Graph
Answer the following questions about the function W graphed in Figure 1.
(a) What was Mr. Hector’s weight at age 10? At age 30?
(b) Did his weight increase or decrease between the ages of 40 and 50? Between
the ages of 50 and 70?
(c) How did his weight change between the ages of 20 and 30?
(d) What was his minimum weight between the ages of 30 and 50?
(e) What was his maximum weight between the ages of 30 and 70?
(f) What is the net change in his weight from the age of 30 to 50?
Solution
(a) His weight at age 10 is W(10). The value of W(10) is the height of the graph
above the x-value 10. From the graph we see that W 1102 = 80. Similarly,
from the graph, W 1302 = 150.
(b) From the graph we see that the values of the function W were increasing
between the x-values 40 and 50, so Mr. Hector’s weight was increasing during
that period. However, the graph indicates that his weight was decreasing
between the ages of 50 and 70.
(c) From the graph we see that Mr. Hector’s weight was constant between the
ages of 20 and 30. He maintained his weight at 150 lb during that period.
(d) From the graph we see that the minimum value that W achieves between the
x-values of 30 and 50 is 130. So Mr. Hector’s minimum weight during that
period was 130 lb.
(e) From the graph we see that the maximum value that W achieves between the
ages of 30 and 70 is 200 lbs. So Mr. Hector’s maximum weight during that
period was 200 lb.
(f) From the graph we see that at age 30 Mr. Hector weighed 150 lb and at age 50
he weighed 200 lb. We have W1502 - W1302 = 200 - 150 = 50, so the net
change in his weight between those two ages is 50 lb.
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NOW TRY EXERCISE 41
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A complete graph of a function contains all the information about the function,
because the graph tells us which input values correspond to which output values. To
analyze the graph of a function, we must keep in mind that the height of the graph is
the value of the function. So we can read the values of a function from its graph.
e x a m p l e 2 Finding the Values of a Function from a Graph
T (*F)
40
30
20
10
0
1
figure 2
2
3
4
5
6
x
The function T graphed in Figure 2 gives the temperature between noon and 6:00 P.M.
at a certain weather station.
(a) Find T(1), T(3), and T(5).
(b) Which is larger, T(2) or T(4)?
(c) Find the value(s) of x for which T1x2 = 25.
(d) Find the values of x for which T1x2 Ú 25.
(e) Find the net change in temperature between 3:00 and 5:00 P.M.
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Data, Functions, and Models
Solution
(a) T 11 2 is the temperature at 1:00 P.M. It is represented by the height of the graph
above the x-axis at the x-value 1. Thus T 112 = 25. Similarly, T 132 = 30 and
T 15 2 = 20.
(b) Since the graph is higher at the x-value 2 than at the x-value 4, it follows that
T12 2 is greater than T142 .
(c) The height of the graph is 25 when x is 1 and when x is 4. So T 1x2 = 25 when
x is 1 and when x is 4.
(d) The graph is higher than 25 for x between 1 and 4. So T 1x2 Ú 25 for all
x-values in the interval [1,4].
(e) From the graph we know that T 152 is 20 and T 132 is 30. We have
Interval notation is reviewed
in Algebra Toolkit A.2,
page T7.
T152 - T132 = 20 - 30
= - 10
So the net change in temperature is - 10°F.
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NOW TRY EXERCISES 7 AND 43
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e x a m p l e 3 Where Graphs of Functions Meet
Use a graphing calculator to draw graphs of the functions f 1x2 = 5 - x 2 and
g1x2 = 3 - x in the same viewing rectangle.
(a) Find the value(s) of x for which f 1x2 = g1x2 .
(b) Find the values of x for which f 1x 2 Ú g1x2 .
(c) Find the values of x for which f 1x 2 6 g1x2 .
Solution
We graph the equations
6
y1 = 5 - x 2
4
_3
_1
f i g u r e 3 Graphs of f and g
and
y2 = 3 - x
in the same viewing rectangle in Figure 3.
(a) Recall that the value of a function is the height of the graph. So f 1x2 = g1x2 at
the x-values where the graphs of f and g meet. From Figure 3 we see that the
graphs meet when x is - 1 and when x is 2. So f 1x 2 = g1x2 when x is - 1 and
when x is 2.
(b) We need to find the x-values where f 1x2 Ú g1x2 . These are the x-values where
the graph of f is above the graph of g. From Figure 3 we see that this happens
for x between - 1 and 2. So f 1x2 Ú g1x 2 for x in the interval 3- 1, 24 .
(c) We need to find the x-values where f 1x2 6 g1x2 . These are the x-values where
the graph of f is below the graph of g. From Figure 3 we see that this happens
for x strictly less than - 1 and x strictly bigger than 2, that is, x 6 - 1 and
x 7 2. (We don’t include the points - 1 and 2 because of the strict inequality.)
So f 1x2 6 g1x2 for x in the intervals 1- q, - 12 and 12, q 2 .
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NOW TRY EXERCISE 11
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SECTION 1.7
2
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Working with Functions: Getting Information from the Graph
77
■ Domain and Range from a Graph
Recall from Section 1.4 that for a function of the form y = f 1x2 we have the following:
Domain
Range
Inputs
Independent variable
x-values
Outputs
Dependent variable
y-values
So since the graph of f consists of the ordered pairs 1x, y2 , the domain and range of
the function can be obtained from the graph as follows.
Domain and Range from a Graph
The domain and range of a function f are represented in the graph of the
function as shown in the figure.
y
f
Range
0
x
Domain
For the function W graphed in Figure 1 on page 74, the domain is the interval
[0, 70] and the range is the interval [10, 200]. Note that the domain consists of all inputs (ages of Mr. Hector) and the range consists of all outputs (weights of Mr. Hector).
e x a m p l e 4 Domain and Range from a Graph
Let f be the function defined by f 1x2 = 24 - x 2.
(a) Use a graphing calculator to draw a graph of f.
(b) Find the domain and range of f from the graph.
Solution
(a) The graph is shown in Figure 4.
Range=[0, 2]
_2
0
Domain=[_2, 2]
figure 4
2
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Data, Functions, and Models
(b) From the graph we see that the domain is the interval 3- 2, 24 and the range is
the interval 30, 24 .
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2
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NOW TRY EXERCISE 19
■ Increasing and Decreasing Functions
At the beginning of this section we saw that the graph of Mr. Hector’s weight rises
when his weight increases and falls when his weight decreases. In general, a function is said to be increasing when its graph rises and decreasing when its graph falls.
Increasing and Decreasing Functions
■
■
The function f is increasing if the values of f 1x2 increase as x increases.
That is, f is increasing on an interval I if f 1a 2 6 f 1b2 whenever a 6 b in I.
The function f is decreasing if the values of f 1x2 decrease as x increases.
That is, f is decreasing on an interval I if f 1a2 7 f 1b2 whenever a 6 b in I.
y
f is decreasing
f is increasing
f
f is increasing
0
x
e x a m p l e 5 Increasing and Decreasing Functions
Use a graphing calculator to draw a graph of f 1x2 = x 3 - 3x 2 + 2.
(a) Find the intervals on which f is increasing.
(b) Find the intervals on which f is decreasing.
Solution
Using a graphing calculator, we draw a graph of the function f as shown in Figure 5.
(a) From the graph we see that f is increasing on 1- q, 04 and on 32, q 2 (represented in red in Figure 6).
5
5
4
_2
_5
_5
figure 5
4
_2
figure 6
SECTION 1.7
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Working with Functions: Getting Information from the Graph
79
(b) From the graph we see that f is decreasing on 30, 24 (represented in blue in
Figure 6).
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2
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NOW TRY EXERCISE 27
■ Local Maximum and Minimum Values
Finding the largest or smallest values of a function is important in many applications.
For example, if a function represents profit, then we are interested in its maximum
value. For a function that represents cost, we would be interested in its minimum
value. We can find these values from the graph of a function. We first define what we
mean by a local maximum or minimum.
Local Maximum and Minimum Values
■
■
The function value f 1a2 is a local maximum value of f if
f 1a2 Ú f 1x2
for values of x near a
f 1a2 … f 1x2
for values of x near a
The function value f 1a2 is a local minimum value of f if
y
Local maximum
value f(a)
f
Local minimum
value f(b)
0
Intervals are studied in Algebra
Toolkit A.2, page T7.
a
b
x
The statement “f 1a2 Ú f 1x2 for values of x near a” means that f 1a2 Ú f 1x2 for
all x in some open interval containing a. Similarly, the statement “f 1a2 … f 1x2 for
values of x near a” means that f 1a2 … f 1x2 for all x in some open interval containing a.
e x a m p l e 6 Local Maximum and Minimum Values of Functions
Find the local maximum and minimum values of f 1x2 = x 3 - 3x 2 + 2.
Solution
The graph of f is shown in Figure 7 on the next page. From the graph we see that f has
a local maximum value 2 at the x-value 0. In other words, f 10 2 = 2 is a local maximum
CHAPTER 1
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Data, Functions, and Models
value (represented by the red dot on the graph in Figure 7). Similarly, f 122 = - 2 is a
local minimum value (represented by the blue dot on the graph in Figure 8).
5
5
4
_2
4
_2
_5
_5
figure 7
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IN CONTEXT ➤
Manfred Steinbach/Shutterstock.com 2009
80
figure 8
NOW TRY EXERCISE 37
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Highway engineers use mathematics to study traffic patterns and relate them to
different road conditions. One feature that they are interested in is the carrying capacity of a road—that is, the maximum number of cars that can safely travel along a
certain stretch of highway. If the cars drive very slowly past a given point on the road,
only a few can pass by every minute. On the other hand, if the cars are zooming
quickly past that point, safety concerns require them to be spaced much farther apart,
so again not many can pass by every minute. Between these two extremes is an optimal speed at which these two competing tendencies balance to allow as many cars
as possible to drive down this stretch of road.
In the next example we use a graphing calculator to analyze the graph of a function developed by an engineer to model the carrying capacity of a highway. (See
Exploration 3 on page 560 to learn how this model is obtained.) The model assumes
that all drivers observe the “safe following distance” guidelines; in reality, the majority of drivers do not, resulting in traffic congestion and accidents.
e x a m p l e 7 Highway Engineering
A highway engineer develops a formula to estimate the number of cars that can
safely travel a particular highway at a given speed. She assumes that each car is
17 feet long, travels at a speed of x mi/h, and follows the car in front of it at the
safe following distance for that speed. She finds that the number N of cars that
can pass a given point per minute is modeled by the function
N 1x2 =
88x
17 + 17 a
x 2
b
20
Graph the function in the viewing rectangle [0, 100] by [0, 60].
(a) Where is the function N increasing? Decreasing?
(b) What is the local maximum value of N? At what x-value does this local
maximum occur?
(c) At what speed is the maximum carrying capacity of the road achieved?
SECTION 1.7
100
f i g u r e 9 Highway capacity at
speed x
Working with Functions: Getting Information from the Graph
81
Solution
60
0
■
The graph is shown in Figure 9.
(a) From the graph we see that the function N is increasing on 30, 204 and
decreasing on 320, 1004 .
(b) From the graph we see that N has a local maximum value of about 52 at the
x-value 20. So the highway can accommodate more cars at about 20 mi/h than
at higher or lower speeds.
(c) Since N has a local maximum value at the x-value 20, the maximum carrying
capacity of the road is achieved at 20 mi/h.
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NOW TRY EXERCISE 49
Test your skill in working with interval notation. Review this topic in Algebra
Toolkit A.2 on page T7.
1–4 A set of numbers is given.
(a) Give a verbal description of the set.
(b) Express the set in interval notation.
(c) Graph the set on the number line.
1. 5x Έ 1 … x … 46
2. 5x Έ - 3 … x 6 26
3. 5x Έ - 10 6 x 6 - 36
4. 5x Έ x Ú 06
5–8 An interval is given.
(a) Give a verbal description of the interval.
(b) Express the interval in set-builder notation.
(c) Graph the interval on the number line.
6. 1- 4, 34
5. (2, 6)
7. 1- q, 22
8. 3- 1, q 2
9–12 The graph of an interval is given.
(a) Give a verbal description of the interval.
(b) Express the interval in set-builder notation.
(c) Express the graphed interval in interval notation.
9.
1
5
11.
10.
_1
1
_3
0
12.
_2
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Data, Functions, and Models
1.7 Exercises
CONCEPTS
Fundamentals
1–4
■
These exercises refer to the graph of the function f shown at the left.
1. To find a function value f 1a2 from the graph of f, we find the height of the graph above
y
the x-axis at x ϭ _______. From the graph of f we see that f 132 = _______.
2. The domain of the function f is all the _______-values of the points on its graph, and
the range is all the corresponding _______-values. From the graph we see that the
f
domain of f is the interval _______ and the range of f is the interval _______.
3
0
3. (a) If f is increasing on an interval, then the y-values of the points on the graph
3
x
_______ (increase/decrease) as the x-values increase. From the graph we see that f
is increasing on the intervals _______ and _______.
(b) If f is decreasing on an interval, then the y-values of the points on the graph
_______ (increase/decrease) as the x-values increase. From the graph we see that f
is decreasing on the intervals _______ and _______.
4. (a) A function value f 1a2 is a local maximum value of f if f 1a2 is the _______ value
of f on some interval containing a. From the graph we see that one local maximum
value of f is _______ and that this value occurs when x is _______.
(b) A function value f 1a2 is a local minimum value of f if f 1a2 is the _______ value
of f on some interval containing a. From the graph we see that one local minimum
value of f is _______ and that this value occurs when x is _______.
Think About It
5. In Example 7 we saw a real-world situation in which it is important to find the
maximum value of a function. Name several other everyday situations in which a
maximum or minimum is important.
6. Draw a graph of a function f that is defined for all real numbers and that satisfies the
following conditions: f is always decreasing and f 1x2 7 0 for all x.
SKILLS
7. The graph of a function h is given.
(a) Find h 1- 2 2, h 102, h 122 , and h 13 2 .
(b) Find the domain and range of h.
(c) Find the values of x for which h 1x2 = 3.
(d) Find the values of x for which h 1x2 … 3.
(e) Find the net change in the value of h when
x changes from - 2 to 4.
8. The graph of a function g is given.
(a) Find g1- 22, g102 , and g172 .
(b) Find the domain and range of g.
(c) Find the values of x for which g1x2 = 4.
(d) Find the values of x for which g1x2 7 4.
(e) Find the net change in the value of g when
x changes from 2 to 7.
y
3
_3
h
0
3
x
y
4
0
g
4
x
SECTION 1.7
■
Working with Functions: Getting Information from the Graph
83
y
9. The graph of a function g is given.
(a) Find g1- 42 , g1- 22 , g10 2 , g122 , and g14 2 .
(b) Find the domain and range of g.
3
g
0
_3
3
x
y
10. Graphs of the functions f and g are given.
(a) Which is larger, f 102 or g10 2 ?
(b) Which is larger, f 1- 32 or g1- 3 2 ?
(c) For which values of x is f 1x2 = g1x 2 ?
g
f
2
0
_2
2
x
_2
■
Graph the functions f and g with a graphing calculator. Use the graphs to find the
indicated values or intervals; state your answer correct to two decimal places.
(a) Find the value(s) of x for which f 1x2 = g1x 2 .
(b) Find the values of x for which f 1x2 Ú g1x2 .
(c) Find the values of x for which f 1x 2 6 g1x 2 .
11–14
11. f 1x2 = x 2 - 5x + 1, g1x 2 = - 3x + 4
12. f 1x2 = - 2x 2 + 3x - 1, g1x2 = 3x - 9
13. f 1x2 = 2x 2 + 3, g1x2 = - x 2 + 3x + 5
14. f 1x2 = 1 - x 2,
g1x2 = x 2 - 2x - 1
15–22 ■ A function f is given.
(a) Use a graphing calculator to draw the graph of f.
(b) Find the domain and range of f from the graph.
15. f 1x2 = x - 1
16. f 1x2 = 4
19. f 1x2 = 216 - x 2
20. f 1x2 = - 225 - x 2
17. f 1x2 = - x 2
18. f 1x2 = 4 - x 2
21. f 1x2 = 1x - 1
23–26
23.
■
22. f 1x2 = 1x + 2
The graph of a function is given. Determine the intervals on which the function is
(a) increasing and (b) decreasing.
y
24.
y
1
1
0
1
1
x
x
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Data, Functions, and Models
25.
y
26.
y
1
1
0
1
x
x
1
27–32 ■ A function f is given.
(a) Use a graphing device to draw the graph of f.
(b) State approximately the intervals on which f is increasing and on which f is
decreasing.
27. f 1x 2 = x 2 - 5x
28. f 1x2 = x 3 - 4x
31. f 1x 2 = x 3 + 2x 2 - x - 2
32. f 1x2 = x 4 - 4x 3 + 2x 2 + 4x - 3
30. f 1x2 = x 4 - 16x 2
29. f 1x 2 = 2x 3 - 3x 2 - 12x
33–36 ■ The graph of a function is given.
(a) Find all the local maximum and minimum values of the function and the value of x
at which each occurs.
(b) Find the intervals on which the function is increasing and on which the function is
decreasing.
y
33.
34.
1
1
0
35.
y
1
x
y
36.
1
x
1
x
y
1
1
1
x
0
37–40 ■ A function is given. Use a graphing calculator to draw a graph of the function.
(a) Find all the local maximum and minimum values of the function and the value of x
at which each occurs. State each answer correct to two decimals.
(b) Find the intervals on which the function is increasing and on which the function is
decreasing. State each answer correct to two decimal places.
37. f 1x 2 = x 3 - x
39. F 1x 2 = x16 - x
38. g1x 2 = 3 + x + x 2 - x 3
40. G 1x2 = x 2x - x 2
SECTION 1.7
CONTEXTS
■
Working with Functions: Getting Information from the Graph
85
41. Power Consumption The figure shows the power consumption in San Francisco for
September 19, 1996. (P is measured in megawatts; t is measured in hours starting at
midnight.)
(a) What was the power consumption at 6:00 A.M.? At 6:00 P.M.?
(b) When was the power consumption a maximum?
(c) When was the power consumption a minimum?
(d) What is the net change in the values of P as the value of x changes from 0 to 12?
P (MW)
800
600
400
200
0
3
6
9
12
15
18
21
t (h)
Source: Pacific Gas & Electric.
42. Earthquake The graph shows the vertical acceleration of the ground from the 1994
Northridge earthquake in Los Angeles, as measured by a seismograph. (Here t
represents the time in seconds.)
(a) At what time t did the earthquake first make noticeable movements of the earth?
(b) At what time t did the earthquake seem to end?
(c) At what time t was the maximum intensity of the earthquake reached?
(d) What is the approximate net change in the intensity of the earthquake as the value
of t changes from 5 to 30?
a (cm/s2)
100
50
−50
5
10 15 20 25 30 t (s)
Source: California Department of Mines and Geology.
43. Low Temperatures In January 2007 the state of California experienced remarkably
cold weather. Many crops that usually thrive in California were lost because of the frost.
Orange crops just ripening in Tulare County, California, were frozen on the trees. The
table and graph on the next page show the daily low temperatures T in Tulare County
for the month of January 2007.
(a) Find T 112 and T 1152 .
(b) Which is larger, T 1152 or T 1192 ?
(c) On what day(s) was the daily low temperature below 32°F? On what day was it the
lowest?
(d) Find the net change in the daily low temperatures from January 1 to January 31.