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. 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) = 3x 2 – 5 b. f(x) = 3x 2 – 5 Solution a. 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 x 2. 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. Solution For 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. x0123 f(x)f(x)

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example Sketch the graph of f(x) = x – 3. Use the graph to evaluate f(4). Solution Begin by creating a table. Plot the points and sketch a line through the points. xy

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) Sketch the graph of f(x) = x – 3. Use the graph to evaluate f(4). Solution To evaluate f(4), first find x = 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. Solution Total 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)0123 Temperature102°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 or f(x) = 102. Elapsed Time (hours)0123 Temperature102°F

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Example (cont) c. Sketch a graph of f together with the data. Elapsed Time (hours)0123 Temperature102°F