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Functions
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2. 2.5 Linear Functions and Models
Copyright © Cengage Learning. All rights reserved.

3. Objectives Linear Functions Slope and Rate of Change
Making and Using Linear Models

4. Linear Functions

5. Linear Functions
Note that a linear function is a function of the form f (x) = ax + b. So in the expression defining a linear function the variable occurs to the first power only. We can also express a linear function in equation form as y = ax + b.

6. Example 1 – Identifying Linear Functions
Determine whether the given function is linear. If the function is linear, express the function in the form f (x) = ax + b.
(a) f (x) = 2 + 3x (b) g(x) = 3(1 – 2x)
(c) h(x) = x(4 + 3x) (d) k(x) =

7. Example 1 – Solution (a) We have f (x) = 2 + 3x = 3x + 2.
So f is a linear function in which a is 3 and b is 2.
(b) We have
g(x) = 3(1 – 2x)
= – 6x + 3.
So g is a linear function in which a is – 6 and b is 3.

8. Example 1 – Solution (c) We have h(x) = x(4 + 3x) = 4x + 3x2,
cont’d
(c) We have
h(x) = x(4 + 3x)
= 4x + 3x2,
which is not a linear function because the variable x is squared in the second term of the expression for h.
(d) We have
k(x) = =
So k is a linear function in which a is and b is

9. Slope and Rate of Change

10. Slope and Rate of Change
Let f (x) = ax + b be a linear function.
If x1 and x2 are two different values for x and if y1 = f (x1) and y2 = f (x2), then the points (x1, y1) and (x2, y2) lie on the graph of f.

11. Slope and Rate of Change
From the definitions of slope and average rate of change we have
We know that the slope of a linear function is the same between any two points.
From the above equation we conclude that the average rate of change of a linear function is the same between any two points. Moreover, the average rate of change is equal to the slope.

12. Slope and Rate of Change
Since the average rate of change of a linear function is the same between any two points, it is simply called the rate of
change.

13. Example 3 – Slope and Rate of Change
A dam is built on a river to create a reservoir. The water level f (t) in the reservoir at time t is given by
f (t) = 4.5t where t is the number of years since the dam was constructed and f (t) is measured in feet.
(a) Sketch a graph of f.
(b) What is the slope of the graph?
(c) At what rate is the water level in the reservoir changing?

14. Example 3 – Solution (a) A graph of f is shown in Figure 2.
Water level as a function of time
Figure 2

15. Example 3 – Solution
cont’d
(b) The graph is a line with slope 4.5, the coefficient of t.
(c) The rate of change of f is 4.5, the coefficient of t. Since time t is measured in years and the water level f (t) is measured in feet, the water level in the reservoir is changing at the rate of 4.5 ft per year. Since this rate of change is positive, the water level is rising.

16. Making and Using Linear Models

17. Making and Using Linear Models
When a linear function is used to model the relationship between two quantities, the slope of the graph of the function is the rate of change of the one quantity with respect to the other. For example, the graph in Figure 3(a) gives the amount of gas in a tank that is being filled. The slope between the indicated points is
(a) Tank filled at 2 gal/min Slope of line is 2
Amount of gas as a function of time
Figure 3

18. Making and Using Linear Models
The slope is the rate at which the tank is being filled, 2 gal per minute. In Figure 3(b) the tank is being drained at the rate of 0.03 gal per minute, and the slope is – 0.03.
(b) Tank drained at 0.03 gal/min Slope of line is – 0.03
Amount of gas as a function of time
Figure 3

19. Example 4 – Making a Linear Model from a Rate of Change
Water is being pumped into a swimming pool at the rate of 5 gal per min. Initially, the pool contains 200 gal of water.
(a) Find a linear function V that models the volume of water in the pool at any time t.
(b) If the pool has a capacity of 600 gal, how long does it take to completely fill the pool?
There are 200 gallons of water
In the pool at time t = 0.

20. Example 4 – Solution (a) We need to find a linear function
V(t) = at + b
that models the volume V(t) of water in the pool after t minutes. The rate of change of volume is 5 gal per min, so a = 5.
Since the pool contains 200 gal to begin with, we have V(0) = a  0 + b = 200, so b = 200.
Now that we know a and b, we get the model
V(t) = 5t + 200

21. Example 4 – Solution
cont’d
(b) We want to find the time t at which V(t) = So we need to solve the equation
600 = 5t + 200
Solving for t, we get
t = So it takes 80 min to fill the pool.