A linear function is a function that has a constant rate of change. Its graph is a straight line.
In an earlier lesson, you learned that the slope of a line is a ratio that compares a vertical change to the corresponding horizontal change.
slopeequals . fraction verticalchange , over horizontalchange end fraction
You can also describe slope as a rate of change. A rate of change is a comparison between two quantities that are changing. If you draw a graph of the quantities, then
rateofchangeequals . fraction verticalchange , over horizontalchange end fraction
Consider the following set of ordered pairs.
{(2, 1), (5, 3), (8, 5), (11, 7)}
Using the vertical line test, you can see that the graph represents a function. Since the graph is a straight line, the ordered pairs represent a linear function.
table with 2 rows and 1 column , row1 column 1 , rateofchangeequals . fraction verticalchange , over horizontalchange end fraction , row2 column 1 , equals , 2 thirds , end table
Example Graphing Ordered Pairs to Identify Linear Functions
Graph each set of ordered pairs to determine if it represents a linear function. If it does, find the rate of change.
a. {(1, 5), (2, 7), (3, 9), (4, 11), (5, 13)}
b. {(0, 0), (1, 1), (2, 4), (3, 9)}