Lesson 6: Connect Proportional Relationships and Slope

Property

The slope of a line is a rate of change that measures the steepness of the line.

It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate. In symbols:

Examples

• A line passes through the points $(3, 5)$ and $(7, 13)$. Its slope is $m = \frac{13-5}{7-3} = \frac{8}{4} = 2$.
• For a line containing points $(-2, 9)$ and $(4, 6)$, the slope is $m = \frac{6-9}{4-(-2)} = \frac{-3}{6} = -\frac{1}{2}$.
• The slope of a line passing through $(100, 50)$ and $(120, 40)$ is $m = \frac{40-50}{120-100} = \frac{-10}{20} = -0.5$.

Property

To find the slope of a line from its graph:

1. Locate two points on the graph whose coordinates are integers.
2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
3. Count the rise and the run on the legs of the triangle.
4. Take the ratio of rise to run to find the slope, $m = \frac{\operatorname{rise}}{\operatorname{run}}$.

Examples

• A line passes through $(1, 2)$ and $(4, 8)$. The rise is $8 - 2 = 6$ and the run is $4 - 1 = 3$. The slope is $m = \frac{6}{3} = 2$.
• A line passes through $(0, 6)$ and $(2, 2)$. The rise is $2 - 6 = -4$ and the run is $2 - 0 = 2$. The slope is $m = \frac{-4}{2} = -2$.
• A line on a graph connects points $(-3, 1)$ and $(5, 7)$. The rise is $7-1=6$ and the run is $5-(-3)=8$. The slope is $m = \frac{6}{8} = \frac{3}{4}$.

Explanation

To find slope from a graph, pick two easy-to-read points. Count the vertical distance (rise) and horizontal distance (run) to get from one point to the other. The slope is simply the rise divided by the run.