Upload
aaliyah-templeton
View
216
Download
0
Tags:
Embed Size (px)
Citation preview
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Section 2.2
Linear Functions
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Objectives
• Basic Concepts
• Representations of Linear Functions
• Modeling Data with Linear Functions
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
, 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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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.
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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.
Elapsed Time (hours) 0 1 2 3
Temperature 102°F 102°F 102°F 102°F
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Example (cont)
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
0
20
40
60
80
100
120
0 1 2 3
Time (hours)
Te
mp
era
ture
(d
eg
ree
s)