Linear Functions and Their PropertiesSection 4.1

Linear FunctionsA linear function can be written as

is the slope and is the -intercept.Otherwise, the function is said to be nonlinear.

Average Rate of Change of a Linear FunctionFor a linear function, the average rate of change is a constant.The average rate of change from to is

How does the numerator simplify?

Increasing, Decreasing or Constant• If , the graph is decreasing on the interval .• If , the graph is constant on the interval .• If , the graph is increasing on the interval .

ExampleSuppose a linear function is written as

Write the function in the form . In terms of , and , what are and ?

ExamplesFor each of the following, • Determine and .• Determine if the graph is increasing, decreasing or constant.1. 2. 3.

ExampleOnly linear functions have a constant rate of change. So, if the average rate of change is not constant, we can conclude the function is not linear.

Below are points for a function. Is it a linear function?2.1 10.98.3 34.812.4 52.1

Exercise 49 – Page 287A truck rental company rents a truck for one day by charging $29 plus $0.07 per mile.a) Write a linear model that relates the cost , in dollars, of renting the truck to the number of miles driven, .b) What is the cost of renting the truck if the truck is driven 110 miles? 230 miles?

Exercise 22 – Page 322A customer must decide between two long-distance phone companies.• Company A charges a monthly fee of $7.00 plus 6 cents per minute.• Company B does not charge a monthly fee but rather charges 8 cents per minute.a) For each company, find a linear function that relates cost, , to the total minutes on the phone, .

Exercise 22 – Page 322 - continuedb) Determine the number of minutes, , for which the monthly bill from Company A will equal the monthly bill from Company B.c) Over what interval of minutes will the bill from Company B be less than the bill from Company A?