A linear function is an expression that when graphed, results in a straight line. The name is slightly revealing—when something is “linear”, it means it progresses in a straight line. Here, you will learn how to recognize, and draw graphs of, linear functions.
A linear function can be written in the form
where is the slope, and is the -intercept, the place the graph intersects with the -axis (and where ).
You can find the slope of a line if you have the coordinates of two points on that line. Call the points and . You use the following formulas for the slope and the constant term :
The straight line that goes through the points and has the slope
and the constant term
The slope tells you how much increases or decreases by as increases by 1.
If , the graph rises towards the right, meaning is increasing as increases. If , the graph sinks towards the right, meaning is decreasing as increases.
The graph intersects the -axis at the point , which is why it is known as the -intercept.
The graph is a straight line that with coordinates .
Find the slope of the straight line that passes through the points and , and find the -intercept.
You set equal to and equal to . (The calculations would still work even if you switched the points.) You get:
You now know that decreases by 2 when increases by 1. In other words, the graph slopes downward by 2 when it moves 1 to the right.
Let’s see what the -intercept is:
Thus, the point -intercept is .