Linear Functions

A function f is a linear function if, for real numbers a and b, $f\left(x\right)=ax+b$.

If $a\ne0,\:$then the domain and the range of $f\left(x\right)\:$are both $\mathbb{R},\:$or in interval notation $\left(-\infty,\infty\right).$

If $a=0,\:$then the function is of the form $f\left(x\right)=b,\:$which represents a horizontal line with y-intercept at $\left(0,b\right).$The domain of this function would be $\left(-\infty,\infty\right),\:$but it's range would be $\lbrace b\rbrace.$

Graphing Linear Functions

In order to graph a line, one must only graph two points on it and then extend the line segment connecting these two points in both directions. There are two basic methods that are frequently used to do this. These are:

1. The Intercept Method
2. The Point and Slope Method

Standard Form

The Intercept Method is the method that is easiest to use when a line is in Standard or General Form, which looks like the following:

$Ax+By=C$.

Intercept Method

To use the intercept method, one finds the x and y intercepts of the line by setting x equal to 0, and then setting y equal to zero.  Once we have obtained the x and y intercepts, we graph these two points, and then the line. Here is an example:

Example

Use the intercept method for graphing the line represented by $-2x+y=8.\:$Try this on your own, and then watch the video below to see the solution.

Solution