6.06 Slope-intercept form

Here is a quick recap of what we know about straight lines on the coordinate plane:

• The slope is a measure of the steepness of a line
• An increasing line has a positive slope
• A decreasing line has a negative slope
• Slope can be determined using $$rise run ​
• A horizontal line has a slope of zero
• The slope of a vertical line is undefined
• The $$x-intercept is the point where the line crosses the $$x-axis
• The $$y-intercept is the point where the line crosses the $$y-axis

Slope-intercept form of a straight line

Any straight line on the coordinate plane is defined entirely by its slope and its $$y-intercept. These two features are all we need to write an equation for the line.

The variable $$a is used to represent the slope, and the variable $$b is used for the $$y-intercept.

We can represent the equation of any straight line, except vertical lines, using what is known as the slope-intercept form of a straight line.

To represent a particular line on the coordinate plane, with it's own unique slope and $$y-intercept, we simply replace $$a and $$b with their corresponding values.

We can use the applet below to see the effect of varying $$a and $$b on both the line and its equation.

As can be seen from the applet, we can make the following statements about the slope.

Slope

The value of the slope, $$a, relates to the line as follows:

• A negative slope ($$a<0) means the line is decreasing
• A positive slope ($$a>0) means the line is increasing
• A zero slope ($$a=0) means the line is horizontal
• The higher the value of $$a, the steeper the line.

Using the slope and $$y-intercept to write the equation of a line

If we know the slope, $$a, and $$y-intercept, $$b, of a line, we can substitute these values into $$y=ax+b to write the equation of the line.

Worked example

Example 1

Write the equation of a line that has a slope of $$34​ and a $$y-intercept of $$−2.

Solution

Substitute $$a=34​ and $$b=−2 into $$y=ax+b

Practice questions

Question 1

Question 2

Use the slope and $$y-intercept to graph a line

To graph a line from an equation, we use the slope, $$a, and the $$y-intercept, $$b.

We begin by locating the $$y-intercept as a point on the $$y-axis.

From this point, we can use the slope to draw the correct slope of the line, as outlined in the three examples below:

Worked example

Example 2

Graph the line that has a slope of $$4 and a $$y-intercept of $$−1.

Solution

By comparing the equation of the line with $$y=ax+b, we see that the slope is $$4 and the y-intercept is $$−1. The slope ($$4=41​) tells us that for a 'run' of $$1, we have a 'rise' of $$4.

We can now create the graph on a coordinate plane, in a series of steps:

• Locate the $$y-intercept at $$−1 on the $$y-axis, and mark it with a point
• From the $$y-intercept, we use the value for the slope to move $$1 unit to the right and then $$4 units up. This gives us the location of a second point.
• Draw a straight line between the two points, and extend the line beyond the points, across the entire coordinate plane.

Example 3

Graph the line with equation $$y=−23​x.

Solution

By comparing the equation of the line with $$y=ax+b, we see that the slope is $$−23​ and the $$y-intercept is $$0, meaning the line passes through the origin. The slope tells us that for a 'run' of $$3, we have a 'rise' of $$−2.

We can now create the graph on a coordinate plane, in a series of steps:

• Locate the $$y-intercept at the origin, $$(0,0), and mark it with a point
• From the $$y-intercept, we use the value for the slope to move $$3 units to the right and then $$2 units down. This gives us the location of a second point.
• Draw a straight line between the two points, and extend the line beyond the points, across the entire coordinate plane.

Practice Question

Question 3

Equation of a horizontal line

A horizontal line has a slope of zero ($$a=0), so the equation of the line becomes:

$$y=b

where $$b is the $$y-intercept of the line.

Here are two examples of horizontal lines:

Horizontal lines $$y=2 and $$y=−3

Equation of a vertical line

A vertical line has an undefined slope, so we can't use the slope-intercept form to write its equation. Instead we define a vertical line as having the equation:

$$x=c

where $$c is the $$x-intercept of the line. 

Here are two examples of vertical lines:

Vertical lines $$x=−1 and $$x=4

Practice question

Question 4