6 Working with Functions: Graphs and Graphing Calculators
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SECTION 1.6
■
Working with Functions: Graphs and Graphing Calculators
65
(b) We make a scatter plot of the data and connect the points with a smooth curve as in
Figure 2(b). Notice that the height of the graph is the value of the function; in other
words, the height of the graph is the temperature of the water at the given time.
T (ºF)
T (ЊF)
100
100
80
60
40
70
5
0
40 x (min)
20
(a) Rough graph
10
20
30
40
50 x (min)
(b) Graph from data
f i g u r e 2 Graph of water temperature as a function of time
■
■
NOW TRY EXERCISE 49
It would be nice to have a precise rule (function) that gives the temperature of
the water at any time. Having such a rule would allow us to predict the temperature
of the water at any time and perhaps warn us about getting scalded. We don’t have
such a rule for this situation, but as we study more functions with different graphs,
we may be able to find a function that models this situation.
2
■ Graphs of Basic Functions
When the rule of a function is given by an equation, then to graph the function, we
first make a table of values. We then plot the points in the table and connect them by
a smooth curve. Let’s try graphing some basic functions.
In the next example we graph a constant function, that is, a function of the form
f 1x2 = c
where c is a fixed constant number. Notice that a constant function has the same output c for every value of the input.
e x a m p l e 2 Graph of a Constant Function
Graph the function f 1x 2 = 3.
Solution
We first make a table of values. Then we plot the points in the table and join them by
a line as in Figure 3. Notice that the graph is a horizontal line 3 units above the x-axis.
■
x
f(x) ؍3
-3
-2
-1
0
1
2
3
3
3
3
3
3
3
3
y
2
0
2
x
f i g u r e 3 Graph of f 1x2 = 3
NOW TRY EXERCISE 7
■
66
CHAPTER 1
■
Data, Functions, and Models
In the next example we graph the function f 1x2 = x. For each input, this function gives the same number as output. This function is called the identity function
(the output is identical to the input). We also graph f 1x2 = x 2, whose graph has the
shape of a parabola.
e x a m p l e 3 Graphs of Basic Functions
Graph the function.
(a) f 1x2 = x
(b) f 1x2 = x 2
(c) f 1x2 = x 3
Solution
We first make a table of values. Then we plot the points in the table and join them by
a line or smooth curve as in Figures 4, 5, and 6.
(a) f 1x2 = x
(b) f 1x2 = x 2
(c) f 1x2 = x 3
x
f 1x2 ؍x
-3
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
3
f 1x2 ؍x2
x
-2
-1
- 12
0
4
1
1
2
1
4
1
2
1
4
1
4
0
x
f 1x2 ؍x3
- 32
-1
- 12
0
- 278
-1
- 18
0
1
2
1
8
1
1
3
2
27
8
y
y
y
2
2
2
0
2
figure 4
0
x
2
figure 5
■
0
x
2
x
figure 6
NOW TRY EXERCISES 15, 19, AND 23
■
Another basic function is the square root function f 1x2 = 1x, which we
graph in the next example.
e x a m p l e 4 Graph of the Square Root Function
Graph the function.
(a) f 1x2 = 1x
(b) g1x 2 = 1x + 3
SECTION 1.6
■
Working with Functions: Graphs and Graphing Calculators
67
Solution
We first make a table of values. Since the domain of f is 5x Έ x Ú 06 , we use only nonnegative values for x. Then we plot the points in the table and join them by a line or
smooth curve as in Figures 7 and 8.
(a) f 1x2 = 1x
(b) g1x 2 = 1x + 3
x
f 1x2 ؍1x
0
0
1
4
1
2
1
2
3
4
5
1
12
13
2
15
Decimal
x
g1x2 ؍1x ؉ 3
Decimal
0
0.5
1.0
1.4
1.7
2.0
2.2
0
0 +3
1
2 + 3
1 +3
12 + 3
13 + 3
2 +3
15 + 3
3.0
3.5
4.0
4.4
4.7
5.0
5.2
1
4
1
2
3
4
5
y
y
2
2
0
2
figure 7
■
0
x
2
x
figure 8
NOW TRY EXERCISE 33
■
In Example 4 notice how the graph of g1x2 = 1x + 3 has the same shape as the
graph of f 1x2 = 1x but is shifted up 3 units. We will study shifting of graphs more
systematically in Chapter 4.
2
■ Graphing with a Graphing Calculator
Algebra Toolkit D.3, page T80,
gives guidelines on using a
graphing calculator as well as
advice on avoiding some
common graphing calculator
pitfalls.
A graphing calculator graphs a function in the same way you do: by making a table of
values and plotting points. Of course, the calculator is fast and accurate; it also frees us
from the many tedious calculations needed to get a good picture of the graph. But the
graph that is produced by a graphing calculator can be misleading. A graphing calculator must be used with care, and the graphs it produces must be interpreted appropriately.
e x a m p l e 5 Graphing a Function
Graph the function f 1x 2 = x 3 - 49x in an appropriate viewing rectangle.
Solution
To graph the function f 1x2 = x 3 - 49x, we first express it in equation form
y = x 3 - 49x
68
CHAPTER 1
■
Data, Functions, and Models
We now graph the equation using a graphing calculator. We experiment with different viewing rectangles. The viewing rectangle in Figure 9(a) gives an incomplete
graph—we need to see more of the graph in the vertical direction. So we choose the
larger viewing rectangle 3- 10, 104 by 3- 200, 2004 by choosing
Xmin = - 10
Ymin = - 200
Xmax = 10
Ymax = 200
The graph in this larger viewing rectangle is shown in Figure 9(b). This graph appears to show all the main features of this function. (We will confirm this when we
study polynomial functions in Chapter 5.)
100
200
_10
10
_10
10
_100
_200
(a)
(b)
f i g u r e 9 Graphing f 1x2 = x 3 - 49x
■
NOW TRY EXERCISE 35
■
e x a m p l e 6 Where Graphs Meet
Consider the functions f 1x 2 = x 2 - 1 and g1x2 = x 3 - 2x - 1.
(a) Graph the functions f and g in the viewing rectangle 3- 2, 34 by 3- 3, 64 .
(b) Find the points where the graphs intersect in this viewing rectangle.
6
Solution
3
_2
The graphs are shown in Figure 10. The graphs appear to meet at three different
points. Zooming in on each point, we find that the points of intersections are
1- 1, 02
_3
f i g u r e 10 Finding where
graphs meet
2
10, - 12
12, 32
You can check that each of these points satisfies both equations.
■
NOW TRY EXERCISE 41
■
■ Graphing Piecewise Defined Functions
Recall that a piecewise defined function is a function that is defined by different rules
on different parts of its domain. As you might expect, the graph of such a function
consists of separate “pieces,” as the following example shows.
SECTION 1.6
■
Working with Functions: Graphs and Graphing Calculators
69
e x a m p l e 7 Graphing a Piecewise Defined Function
Graph the function
f 1x2 = e
x
x+1
if x … 2
if x 7 2
Solution
The rule f is given by f 1x2 = x for x … 2 and f 1x2 = x + 1 for x 7 2. We use this
information to make a table of values and plot the graph in Figure 11.
On many graphing calculators the
graph in Figure 11 can be produced
by using the logical functions in the
calculator. For example, on the
TI-83 the following equation gives
the required graph:
x
f 1x 2
-2
-1
0
1
2
3
4
5
-2
-1
0
1
2
4
5
6
y
6
4
2
_2
0
_1
1
2
3
4
5 x
_2
xՅ2
2Ͻx
f i g u r e 11 Graph of the piecewise defined
Y1=(X◊2)*X+(X>2)*(X+1)
function f
(To avoid the extraneous vertical
line between the two parts of the
graph, put the calculator in Dot
mode.)
Notice the closed and open circles on the graph in Figure 11. The closed circle at the
point (2, 2) indicates that this point is on the graph. The open circle above it at (2, 3)
indicates that this point is not on the graph.
■
NOW TRY EXERCISE 45
■
e x a m p l e 8 Water Rates
A city charges its residents $0.008 per gallon for households that use less than 4000
gallons a month and $0.012 for households that use 4000 gallons or more a month
(see Example 9 in Section 1.5). The function
C 1x 2 = e
y
80
0.008x
0.012x
if x 6 4000
if x Ú 4000
gives the cost of using x gallons of water per month.
(a) Graph the piecewise defined function C.
(b) What does the break in the graph represent?
60
40
Solution
20
0
2000
4000
6000
f i g u r e 12 Graph of the cost
function C
x
(a) We graph the function C in two pieces. For x 6 4000 we graph the equation
y = 0.008x, and for x Ú 4000 we graph the equation y = 0.012x. The graph is
shown in Figure 12.
(b) From the graph we see that at 4000 gallons there is a jump in the cost of water.
This corresponds to the jump in the price from $0.008 to $0.012 per gallon.
■
NOW TRY EXERCISE 55
■
70
CHAPTER 1
■
Data, Functions, and Models
The absolute value function f 1x2 = 0 x 0 is a piecewise defined function:
f 1x2 = e
-x
x
if x … 0
if x 7 0
This function leaves positive inputs unchanged and reverses the sign of negative
inputs.
e x a m p l e 9 Graph of the Absolute Value Function
Sketch the graph of f 1x2 = 0 x 0 .
To graph f 1x 2 = 0 x 0 on the TI-83,
enter the function asY1=abs(X).
Solution
We make a table of values and then sketch the graph in Figure 13.
x
f 1x2 ؍0 x 0
-3
-2
-1
0
1
2
3
3
2
1
0
1
2
3
■
y
2
0
2
x
f i g u r e 13 Graph of f 1x2 = 0 x 0
NOW TRY EXERCISE 31
Test your skill in using your graphing calculator. Review the guidelines on graphing calculators in Algebra Toolkit D.3 on page T80.
1–4 Use a graphing calculator or computer to decide which viewing rectangle
(i)–(iv) produces the most appropriate graph of the equation.
1. y = x 4 + 2
(i)
(ii)
(iii)
(iv)
3- 2, 2 4 by 3- 2, 24 +
[0, 4] by [0, 4]
3- 8, 8 4 by 3- 4, 404
3- 40, 40 4 by 3- 80, 8004
3. y = 10 + 25x - x 3
(i)
(ii)
(iii)
(iv)
3- 4, 4 4 by 3- 4, 44
3- 10, 10 4 by 3- 10, 104
3- 20, 20 4 by 3- 100, 1004
3- 100, 100 4 by 3- 200, 2004
2. y = x 2 + 7x + 6
(i)
(ii)
(iii)
(iv)
3 - 5, 5 4 by 3- 5, 54
[0, 10] by 3- 20, 100 4
3- 15, 8 4 by 3- 20, 1004
3- 10, 3 4 by 3- 100, 204
4. y = 28x - x 2
(i)
(ii)
(iii)
(iv)
3- 4, 4 4 by 3- 4, 44
3- 5, 5 4 by 30, 100 4
3- 10, 10 4 by 3- 10, 404
3- 2, 10 4 by 3- 2, 64
5. Graph the equation in an appropriate viewing rectangle.
(a) y = 50x 2
(c) y = x 4 - 5x 2
(b) y = x 3 - 2x - 3
(d) y = 24 + x - x 2
■
SECTION 1.6
■
Working with Functions: Graphs and Graphing Calculators
71
1.6 Exercises
CONCEPTS
Fundamentals
1. To graph the function f, we plot the points (x, _______ ) in a coordinate plane. To
graph f 1x 2 = x 3 + 2, we plot the points (x, _______ ). So the point (2, _______ ) is
on the graph of f. The height of the graph of f above the x-axis when x = 2 is _______.
2. If f 122 = 3, then the point (2, _______ ) is on the graph of f.
3. If the point (1, 5) is on the graph of f, then f 112 = _______.
4. Match the function with its graph.
(a) f 1x2 = x 2
(b) f 1x2 = x 3
I
II
y
III
y
(d) f 1x2 = 0 x 0
(c) f 1x2 = 1x
IV
y
y
1
0
1
0
x
1
0
x
0
x
1
1
Think About It
5. In what ways can a graph produced by a graphing calculator be misleading? Explain
using an example.
6. A student wishes to graph the following functions on the same screen:
y = x 1>3
y=
and
x
x +4
He enters the following information into the calculator:
Y1ϭX^1>3
and
Y2ϭX>Xϩ4
The calculator graphs two lines instead of the information he wanted. What went
wrong?
SKILLS
7–12
■
A function is given. Complete the table and then graph the function.
7. f 1x2 = 5
x
f(x)
8. g1x 2 = 2x - 4
x
g(x)
9. h 1x2 = 2x 2 - 3
x
-3
-3
-3
-2
-2
-2
-1
-1
-1
0
0
0
1
1
1
2
2
2
3
3
3
h(x)
x
72
CHAPTER 1
■
Data, Functions, and Models
10. k 1x 2 = x 3 + 8
x
11. F 1x 2 = 1x + 4
k(x)
x
12. G 1x2 = 2 0 x + 1 0
F(x)
x
G(x)
-3
-4
-4
-2
-2
-3
-1
0
-2
0
2
-1
1
4
0
2
8
1
3
13–28
■
Sketch the graph of the function by first making a table of values.
13. f 1x2 = 8
15. g1x 2 = x - 4
14. f 1x2 = - 1
16. g1x 2 = 3x - 7
17. h 1x2 = - x + 3, - 3 … x … 3
19. k 1x 2 = - x
20. k 1x 2 = 4 - x
24. G 1x2 = 1x + 22 3
23. G 1x 2 = x 3 - 8
26. A 1x 2 = 1x + 4
25. A 1x 2 = 1x + 1
27. r 1x 2 = 0 2x 0
■
2
22. F 1x 2 = 1x - 32 2
21. F 1x 2 = x - 4
2
29–34
18. h 1x2 = 3 - x, - 2 … x … 2
2
28. r 1x 2 = 0 x + 1 0
Graph the given functions on the same coordinate axes.
29. f 1x2 = x 2,
30. f 1x2 = x ,
2
g1x2 = x 2 - 3, h1x 2 = 14 x 2
g1x2 = 1x + 52 2,
h 1x 2 = 1x - 5 2 2
31. f 1x2 = 0 x 0 , g1x 2 = 2 0 x 0 , h 1x 2 = 0 x + 2 0
32. f 1x2 = 0 x 0 , g1x 2 = 0 3x 0 , h 1x 2 = 0 x 0 + 4
33. f 1x2 = 1x,
34. f 1x2 = 1x,
35–40
■
g1x2 = 14x, h 1x2 = 1x - 4
g1x2 = 1x - 9, h 1x 2 = 1x - 1 + 2
Draw a graph of the function in an appropriate viewing rectangle.
35. f 1x2 = 4 + 6x - x 2
37. f 1x2 = x - 4x
4
39. f 1x2 = `
41–42
■
36. f 1x2 = 212x - 17
38. f 1x2 = 0.1x 3 - x 2 + 1
3
x
+ 7`
2
40. f 1x2 = 2x - 0 x 2 - 5 0
Do the graphs of the two functions intersect in the given viewing rectangle? If
they do, how many points of intersection are there?
41. f 1x2 = 3x 2 + 6x - 12, g1x 2 = 27 42. f 1x2 = 6 - 4x - x ,
2
43–48
■
7
12
g1x 2 = 3x + 18;
x 2;
3- 4, 4 4 by 3- 1, 34
3- 6, 24 by 3- 5, 20 4
Sketch the graph of the piecewise defined function.
43. f 1x2 = e
0
1
if x 6 2
if x Ú 2
44. f 1x2 = e
1
x+1
if x … 1
if x 7 1
SECTION 1.6
CONTEXTS
■
Working with Functions: Graphs and Graphing Calculators
45. f 1x2 = e
1-x
5
if x 6 - 2
if x Ú - 2
46. f 1x2 = e
2x + 3
3-x
47. f 1x2 = e
1 - x2
x
if x … 2
if x 7 2
48. f 1x2 = e
2
x2
73
if x 6 - 1
if x Ú - 1
if x … - 1
if x 7 - 1
49. Filling a Bathtub A bathtub is being filled by a constant stream of water from the
faucet. Sketch a rough graph of the water level in the tub as a function of time.
50. Cooling Pie You place a frozen pie in an oven and bake it for an hour. Then you take
the pie out and let it cool before eating it. Sketch a rough graph of the temperature of
the pie as a function of time.
51. Christmas Card Sales The number of Christmas cards sold by a greeting card store
depends on the time of year. Sketch a rough graph of the number of Christmas cards
sold as a function of the time of year.
52. Height of Grass A home owner mows the lawn every Wednesday afternoon. Sketch a
rough graph of the height of the grass as a function of time over the course of a fourweek period beginning on a Sunday.
r
5
6
7
8
9
10
T(r)
53. Weather Balloon As a weather balloon is inflated, the thickness T of its latex skin is
related to the radius of the balloon by
T 1r 2 =
0.5
r2
where T and r are measured in centimeters. Complete the table in the margin and graph
the function T for values of r between 5 and 10.
54. Gravity near the Moon The gravitational force between the moon and an astronaut in
a space ship located a distance x above the center of the moon is given by the function
F 1x2 =
350
x2
where F is measured in newtons (N) and x is measured in megameters (Mm). Graph the
function F for values of x between 2 and 8.
55. Toll Road Rates The toll charged for driving on a certain stretch of a toll road
depends on the time of day. The amount of the toll charge is given by
5.00
7.00
T 1x2 = e 5.00
7.00
5.00
if 0 … x 6 7
if 7 … x … 10
if 10 6 x 6 16
if 16 … x … 19
if 19 6 x 6 24
where x is the number of hours since 12:00 A.M.
(a) Graph the function T.
(b) What do the breaks in the graph represent?
56. Postage Rates The domestic postage rate depends on the weight of the letter. In 2009,
the domestic postage rate for first-class letters weighing 3.5 oz or less was given by
0.44
0.61
P1x2 = d
0.78
0.95
if 0
if 1
if 2
if 3
…
6
6
6
x
x
x
x
…
…
…
…
where x is the weight of the letter measured in ounces.
(a) Graph the function P.
(b) What do the breaks in the graph represent?
1
2
3
3.5
74
CHAPTER 1
■
Data, Functions, and Models
1.7 Working with Functions:
Getting Information from the Graph
■
Reading the Graph of a Function
■
Domain and Range from a Graph
■
Increasing and Decreasing Functions
■
Local Maximum and Minimum Values
IN THIS SECTION... we use the graph of a function to get information about the
function, including where the values of the function increase or decrease and where the
maximum or minimum value(s) of the function occur.
GET READY... by reviewing interval notation in Algebra Toolkit A.2. Test your skill in
working with interval notation by doing the Algebra Checkpoint at the end of this
section.
The graph of a function allows us to “see” the behavior, or life history, of the function. For example, we can see from the graph of a function the highest or lowest value
of the function or whether the values of the function are rising or falling. So if a function represents cost, the lowest point on its graph tells where the minimum cost occurs. If a function represents profit, its graph can tell us where profit is increasing or
decreasing. In this section we examine how to obtain these and other types of information from the graph of a function.
2
■ Reading the Graph of a Function
If a function models a real-world situation, such as the weight of a person, its graph
is usually easy to interpret. For example, suppose the weight of Mr. Hector (in
pounds) is given by the function W, where the independent variable x is his age in
years. So
W 1x2 = “weight of Mr. Hector at age x”
The graph of the function W in Figure 1 gives a visual representation of how his
weight has changed over time. Note that Mr. Hector’s weight W 1x2 at age x is the
height of the graph above the point x.
W
200
180
160
140
120
100
80
60
40
20
0
W(30)=150
W(10)=80
10
20
30
40
50
60
70 x
f i g u r e 1 Graph of Mr. Hector’s weight
SECTION 1.7
■
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.
■
NOW TRY EXERCISE 41
■
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.