Notes

A *function* is simply a rule that assigns a unique output value to an input value. For example, the function “multiply the input value by 3 and add 1” would assign an output value of 7 to an input value of 2 (since 3 times 2 plus 1 equals 7).

##### Equations

We generally write input values using the variable “*x*” and output values using the variable “*y*.” Then the *equation* representing the previous function is *y* = 3*x* + 1.

##### Tables

One way to summarize a function is in a table of input and output values. A table for the function, *y* = 3*x* + 1, is as follows:

Input, x | 0 | 1 | 2 | 3 | 4 |

Output, y | 1 | 4 | 7 | 10 | 13 |

Graph

Another way to summarize a function is in a graph with the input values (x) plotted on the horizontal axis and the output values (y) plotted on the vertical axis. A graph for the function, *y* = 3*x* + 1, is as follows:

The plotted points on the graph are written in pairs with *x* first and *y* second, e.g., (*x*, *y*) = (1, 4).

##### Linear functions

There are many different types of function, but the most important function in elementary math is the *linear function*. Linear functions have the form *y* = *mx* + *c*, where *m* is the slope and *c* is the y-intercept.

- The slope,
*m*, is the change in*y*÷ the change in*x* - The
*y*-intercept,*c*, is the value of*y*when*x*is 0

In a graph for a linear function, if the slope is positive, the graph slopes up to the right. If the slope is negative, the graph slopes down to the right. The *y*-intercept is where the graph crosses the vertical axis.

For the function, *y* = 3*x* + 1, the slope, *m*, is 3 because *y* changes by 3 when *x* changes by 1. For example, when x changes from 0 to 1 (an increase of 1), y changes from 1 to 4 (an increase of 3). Similarly, when x changes from 1 to 2 (an increase of 1), y changes from 4 to 7 (an increase of 3).

For the function, *y* = 3*x* + 1, the *y*-intercept, *c*, is 1 because *y* is 1 when *x* is 0.

By the way, if the output numbers for this function look familiar, that’s because they came up in Practice Exercise #3 in 8.1: Sequences. Linear functions are related to arithmetic sequences: the step in an arithmetic sequence is equivalent to the slope in a linear function, while the 0th entry in an arithmetic sequence is equivalent to the *y*-intercept in a linear function.

##### Calculating the slope and intercept of a linear function

Consider the linear function represented by the following table and graph:

Input, x | 0 | 1 | 2 | 3 | 4 |

Output, y | –1 | 1 | 3 | 5 | 7 |

- To calculate the slope, m, find the input and output values of two points and calculate the change in
*y*÷ the change in*x*, e.g., (7 – 5) / (4 – 3) = 2 ÷ 1 = 2. - To calculate the
*y*-intercept,*c*, find the output value when the input value is 0, e.g.,*y*= –1 when*x*= 0.

The equation of this linear function is therefore *y* = 2*x* – 1.

The table and graph of a linear function may not show the y-intercept explicitly, in which case the y-intercept might need to be calculated by using the slope and a point. For example, suppose we know the slope is 2 and a point on the graph is (*x*, *y*) = (2, 3). Then, changing *x* by –2 (i.e., from 2 to 0) must change *y* by the slope times the change in *x* = 2 times –2 = –4 (i.e., y changes from 3 to 3 – 4 = –1).

The video below works through an example of a linear function.

Video Tips

Practice Exercises

Do the following exercises to practice working with linear functions.