3.02 Slopes and intercepts

Slopes of lines and axis intercepts

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

• They have a slope (gradient), 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 $x$x-intercepts, $y$y-intercepts or both an $x$x and a $y$y-intercept.
• The slope can be calculated using $\frac{\text{rise }}{\text{run }}$rise run ​ or $\frac{y_2-y_1}{x_2-x_1}$y2​−y1​x2​−x1​​.
• They have an equation of the form $y=a+bx$y=a+bx. (or $y=mx+c)$y=mx+c)

The values of $b$b and $a$a mean specific things. Explore for yourself what these values do by using the following applet. Note that in this applet  $m=b$m=b and $c=a$c=a.

Slope

Using the widget above, it can be observed that the $b$b-value affects the slope of the line. This $b$b-value is actually equal to the slope. So the slope of a line $b=\frac{\text{rise }}{\text{run }}$b=rise run ​ or $\frac{y_2-y_1}{x_2-x_1}$y2​−y1​x2​−x1​​.

• If $b<0$b<0, the slope is negative and the line is decreasing.
• if $b>0$b>0, the slope is positive and the line is increasing.
• if $b=0$b=0 the slope is $0$0 and the line is horizontal.
• The slope is not defined for a straight-line graph that is vertical.
• Also, the larger the value of $b$b the steeper the line.

$y$y-intercept

Using the widget above, it can be observed that the $a$a-value affects the $y$y-axis intercept.  

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

Practice questions

Question 1

Question 2