Graph Linear Functions
In Section 1.4 we discussed lines. In particular, for nonvertical lines we developed the slope–intercept form of the equation of a line y =mx + b. When we write the slope–intercept form of a line using function notation, we have a linear function.
A linear function is a function of the form f1x2 =mx + b
The graph of a linear function is a line with slope m and y-intercept b. Its domain is the set of all real numbers.
Functions that are not linear are said to be nonlinear.
Graphing a Linear Function
Graph the linear function: f1x2 = -3x +7
This is a linear function with slope m = -3 and y-intercept b= 7. To graph this function, we plot the point (0, 7), the y-intercept, and use the slope to find an additional point by moving right 1 unit and down 3 units. See Figure 1.
Alternatively, we could have found an additional point by evaluating the function at some x⬆ 0. For x= 1, we find f112 = -3112 + 7= 4 and obtain the point (1, 4) on the graph.
Now Work P R O B L E M S 1 3 (a) A N D (b)
Use Average Rate of Change to Identify Linear Functions
Look at Table 1, which shows certain values of the independent variable x and corresponding values of the dependent variable y for the function f1x2 = -3x+ 7.
Notice that as the value of the independent variable, x, increases by 1 the value of the dependent variable y decreases by 3. That is, the average rate of change of y with respect to x is a constant, -3.
1
DEFINITION
E X A M P L E 1
Solution
2
Figure 1
x Δy 3 Δx 1 y
1 3 5
1 3 5
(1, 4) (0, 7)
SECTION 3.1 Linear Functions and Their Properties 131
It is not a coincidence that the average rate of change of the linear function f1x2 = -3x+ 7 is the slope of the linear function. That is, ⌬y
⌬x =m = -3. The following theorem states this fact.
Average Rate of Change of a Linear Function
Linear functions have a constant average rate of change. That is, the average rate of change of a linear function f1x2 =mx + b is
⌬y
⌬x = m
Proof The average rate of change of f1x2 = mx+ b from x1 to x2, x1 ⬆ x2, is
⌬y
⌬x = f1x22 - f1x12
x2- x1 = 1mx2+ b2 - 1mx1+ b2 x2 -x1
= mx2 -mx1
x2 -x1 = m(x2- x1) x2- x1 = m
Based on the theorem just proved, the average rate of change of the function g(x) = -2
5 x+ 5 is -2 5 .
Now Work P R O B L E M 1 3 (c)
As it turns out, only linear functions have a constant average rate of change.
Because of this, we can use the average rate of change to determine whether a function is linear or not. This is especially useful if the function is defined by a data set.
Using the Average Rate of Change to Identify Linear Functions (a) A strain of E. coli Beu 397-recA441 is placed into a Petri dish at 30° Celsius
and allowed to grow. The data shown in Table 2 on page 132 are collected. The population is measured in grams and the time in hours. Plot the ordered pairs (x, y) in the Cartesian plane and use the average rate of change to determine whether the function is linear.
THEOREM
E X A M P L E 2
x yⴝf1x2 ⴝ ⴚ3xⴙ7 Average Rate of Changeⴝ ⌬y
⌬x
-2 13
10-13 -1-(-2) = -3
1 = -3
-1 10
7-10 0-(-1) = -3
1 = -3
0 7
-3
1 4
-3
2 1
-3
3 -2
Table 1
(b) The data in Table 3 represent the maximum number of heartbeats that a healthy individual should have during a 15-second interval of time while exercising for different ages. Plot the ordered pairs (x, y) in the Cartesian plane, and use the average rate of change to determine whether the function is linear.
Table 2 Table 3
Time
(hours), x (x, y)
Population (grams), y
0 1 2 3 4 5
(0, 0.09) (1, 0.12) (2, 0.16) (3, 0.22) (4, 0.29) (5, 0.39) 0.09
0.12 0.16 0.22 0.29 0.39
Source:American Heart Association
Age, x (x, y)
Maximum Number of Heartbeats, y 20
30 40 50 60 70
(20, 50) (30, 47.5) (40, 45) (50, 42.5) (60, 40) (70, 37.5) 50
47.5 45 42.5 40 37.5
Compute the average rate of change of each function. If the average rate of change is constant, the function is linear. If the average rate of change is not constant, the function is nonlinear.
(a) Figure 2 shows the points listed in Table 2 plotted in the Cartesian plane. Notice that it is impossible to draw a straight line that contains all the points. Table 4 displays the average rate of change of the population.
Solution
Figure 2
Time (hours), x
Population (grams), y
x y
0.1 0.2 0.3 0.4
5 4 3 2
0 1
Time (hours), x Population (grams), y Average Rate of Change ⴝ ⌬y
⌬x
0 0.09
0.12-0.09 1-0 =0.03
1 0.12
0.04
2 0.16
0.06
3 0.22
0.07
4 0.29
0.10
5 0.39
Table 4
Because the average rate of change is not constant, we know that the function is not linear. In fact, because the average rate of change is increasing as the value of the independent variable increases, the function is increasing at an increasing rate. So not only is the population increasing over time, but it is also growing more rapidly as time passes.
(b) Figure 3 shows the points listed in Table 3 plotted in the Cartesian plane. We can see that the data in Figure 3 lie on a straight line. Table 5 contains the average rate of change of the maximum number of heartbeats. The average rate of change of the heartbeat data is constant, -0.25 beat per year, so the function is linear. We
SECTION 3.1 Linear Functions and Their Properties 133
can find the linear function using the point-slope formula with x1= 20, y1= 50, and m = -0.25.
y- 50= -0.25(x- 20) y- 50= -0.25x+ 5
y= -0.25x+ 55
y- y1= m(x -x1)
Now Work P R O B L E M 2 1
Determine Whether a Linear Function Is Increasing, Decreasing, or Constant
Look back at the Seeing the Concept on page 35. When the slope m of a linear function is positive (m 7 0), the line slants upward from left to right. When the slope m of a linear function is negative (m 6 0), the line slants downward from left to right. When the slope m of a linear function is zero (m= 0), the line is horizontal.
Increasing, Decreasing, and Constant Linear Functions
A linear function f1x2 = mx+ b is increasing over its domain if its slope, m, is positive. It is decreasing over its domain if its slope, m, is negative. It is constant over its domain if its slope, m, is zero.
Determining Whether a Linear Function Is Increasing, Decreasing, or Constant
Determine whether the following linear functions are increasing, decreasing, or constant.
(a) f1x2 = 5x- 2 (b) g(x) = -2x +8
(c) s(t) = 3
4 t- 4 (d) h(z) = 7
(a) For the linear function f1x2 = 5x- 2 , the slope is 5, which is positive. The function f is increasing on the interval (-⬁, ⬁).
(b) For the linear function g1x2 = -2x + 8 , the slope is -2, which is negative. The function g is decreasing on the interval (-⬁, ⬁ ).
(c) For the linear function s(t) = 3
4 t- 4 , the slope is 3
4 , which is positive. The function s is increasing on the interval (-⬁, ⬁).
3
THEOREM
E X A M P L E 3
Solution
Figure 3
Age
Heartbeats
x y
40 45 50
70 60 50 40
20 30
Age, x Maximum Number
of Heartbeats, y Average Rate of Change ⴝ ⌬y
⌬x
20 50
47.5-50 30-20 = -0.25
30 47.5
-0.25
40 45
-0.25
50 42.5
-0.25
60 40
-0.25
70 37.5
Table 5
(d) We can write the linear function h as h(z) =0z+ 7. Because the slope is 0, the function h is constant on the interval (-⬁, ⬁ ).
Now Work P R O B L E M 1 3 (d)
Build Linear Models from Verbal Descriptions
When the average rate of change of a function is constant, we can use a linear function to model the relation between the two variables. For example, if your phone company charges you $0.07 per minute to talk regardless of the number of minutes used, we can model the relation between the cost C and minutes used x as the linear function C(x) = 0.07x, with slope m = 0.07 dollar
1 minute . Modeling with a Linear Function
If the average rate of change of a function is a constant m, a linear function f can be used to model the relation between the two variables as follows:
f1x2 = mx+ b
where b is the value of f at 0, that is, b= f102.
Straight-line Depreciation
Book value is the value of an asset that a company uses to create its balance sheet.
Some companies depreciate their assets using straight-line depreciation so that the value of the asset declines by a fixed amount each year. The amount of the decline depends on the useful life that the company places on the asset. Suppose that a company just purchased a fleet of new cars for its sales force at a cost of $28,000 per car. The company chooses to depreciate each vehicle using the straight-line method over 7 years. This means that each car will depreciate by +28,000
7 = +4000 per year.
(a) Write a linear function that expresses the book value V of each car as a function of its age, x.
(b) Graph the linear function.
(c) What is the book value of each car after 3 years?
(d) Interpret the slope.
(e) When will the book value of each car be $8000?
[Hint: Solve the equation V(x) = 8000.]
(a) If we let V(x) represent the value of each car after x years, then V(0) represents the original value of each car, so V(0)= +28,000. The y-intercept of the linear function is $28,000. Because each car depreciates by $4000 per year, the slope of the linear function is -4000. The linear function that represents the book value V of each car after x years is
V(x) = -4000x +28,000 (b) Figure 4 shows the graph of V.
(c) The book value of each car after 3 years is
V(3)= -4000(3)+ 28,000
= +16,000
(d) Since the slope of V(x) = -4000x+ 28,000 is -4000, the average rate of change of book value is ⫺$4000/year. So for each additional year that passes the book value of the car decreases by $4000.
4
E X A M P L E 4
Solution
Figure 4
28,000 24,000 20,000 16,000 12,000 8000 4000
1 2 3 4 5 6 7
Book value ($)
Age of vehicle (years) x v
SECTION 3.1 Linear Functions and Their Properties 135
(e) To find when the book value will be $8000, solve the equation V(x) = 8000
-4000x+ 28,000= 8000 -4000x= -20,000
x= -20,000 -4000 = 5