3.02 Slopes and intercepts

Here are some key facts about straight lines drawn on the xy-plane.

• They have a gradient (slope) which is a measure of how steep the line is.

• They can be increasing (positive slope) or decreasing (negative slope).

• They can be horizontal (zero slope).

• They can be vertical (slope is undefined).

• They have x-intercepts, y-intercepts or both an x and a y-intercept.

• The slope can be calculated using b=\dfrac{\text{rise}}{\text{run}} or b=\dfrac{y_2-y_1}{x_2-x_1}.

• They have an equation of the form y=a +bx. (or y=mx+c)

The values of b and a have specific meanings.

For the slope:

• If b<0, the slope is negative and the line is decreasing.

• If b>0, the slope is positive and the line is increasing.

• If b=0, the slope is 0 and the line is horizontal.

• The slope is not defined for a straight-line graph that is vertical.

• The larger the value of b the steeper the line.

For the y-intercept:

• If a is positive then the line is vertically translated (moved) up.

• If a is negative then the line is vertically translated (moved) down.

Linear equations have two common forms.

Slope-intercept form: y=a+bx where b is the slope and a is the y-axis intercept.

General form: Ax+By=C where A, \, B, \, C are constants.

To find the slope and y-axis intercept of a straight line equation given in general form, it can rearranged into slope-intercept form.

Examples