Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 2.2 Linear Functions.

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Section 2.2

Linear Functions


Objectives

• Basic Concepts

• Representations of Linear Functions

• Modeling Data with Linear Functions


, 0 and 1,xf x a a a

A function f defined by f(x) = ax + b, where a and b are constants, is a linear function.

LINEAR FUNCTION


Example

Determine whether f is a linear function. If f is a linear function, find values for a and b so that f(x) = ax + b.a. f(x) = 6 – 2x b. f(x) = 3x2 – 5b. f(x) = 3x2 – 5Solutiona. Let a = –2 and b = 6.

Then f(x) = −2x + 6, and f is a linear function. b. Function f is not linear because its formula contains x2. The formula for a linear function cannot contain an x with an exponent other than 1.


Example

Use the table of values to determine whether f(x) could represent a linear function. If f could be linear, write the formula for f in the form f(x) = ax + b.

SolutionFor each unit increase in x, f(x) increases by 7 units so f(x) could be linear with a = 7. Because f(0) = 4, b = 4. thus f(x) = 7x + 4.

x 0 1 2 3

f(x) 4 11 18 25


Example

Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4).SolutionBegin by creating a table.

Plot the points and sketch a line through the points.

x y

−1 −4

0 −3

1 −2

2 −1


Example (cont)

Sketch the graph of f(x) = x – 3 . Use the graph to evaluate f(4).SolutionTo evaluate f(4), first findx = 4 on the x-axis. Then find the corresponding y-value. f(4) = 1


The formula f(x) = mx + b may be interpreted as follows.

f(x) = mx + b

(New amount) = (Change) + (Fixed amount)

When x represents time, change equals (rate of change) × (time).

f(x) = m × x + b

(Future amount) = (Rate of change) × (Time) + (Initial amount)

MODELING DATA WITH A LINEARFUNCTION


Example

Suppose that a moving truck costs $0.25 per mile and a fixed rental fee of $20. Find a formula for a linear function that models the rental fees.SolutionTotal cost is found by multiplying $0.25 (rate per mile) by the number of miles driven x and then adding the fixed rental fee (fixed amount) of $20. Thus f(x) = 0.25x + 20.


Example

The temperature of a hot tub is recorded at regular intervals.

a. Discuss the temperature of the water during this time interval.

b. Find a formula for a function f that models these data.

c. Sketch a graph of f together with the data.

Elapsed Time (hours) 0 1 2 3

Temperature 102°F 102°F 102°F 102°F


Example (cont)

a. Discuss the temperature of the water during this time interval. The temperature appears to be a constant 102°F.

b. Find a formula for a function f that models these data.Because the temperature is constant, the rate of change is 0. Thus f(x) = 0x + 102 or f(x) = 102.




Example (cont)

c. Sketch a graph of f together with the data.



0

20

40

60

80

100

120

0 1 2 3

Time (hours)

Te

mp

era

ture

(d

eg

ree

s)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 2.2 Linear Functions.

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