Lesson 5: Slope

Property

Rate data can be converted to ordered pairs $(time, distance)$ or $(input, output)$ and plotted on a coordinate plane. Lines with steeper slopes represent faster rates of change.

Examples

Property

Given two points in the coordinate plane, $P$ and $Q$, we define the rise to be the difference of the $y$ values from $P$ to $Q$, and the run to be the difference in the $x$ values from $P$ to $Q$. The slope of the line segment is the quotient of these two differences:

If $P$ has the coordinates $(x_0, y_0)$ and $Q$ has the coordinates $(x_1, y_1)$ this is

Examples

• The slope of the line passing through points $P(1, 4)$ and $Q(3, 10)$ is $\operatorname{slope} = \frac{10 - 4}{3 - 1} = \frac{6}{2} = 3$.
• A line passes through $(2, 9)$ and $(5, 3)$. The slope is $\frac{3 - 9}{5 - 2} = \frac{-6}{3} = -2$. The line goes downward as you move to the right.
• For a horizontal line through $(1, 6)$ and $(8, 6)$, the slope is $\frac{6 - 6}{8 - 1} = \frac{0}{7} = 0$.

Explanation

Slope measures a line's steepness and direction. It's the 'rise' (vertical change) divided by the 'run' (horizontal change). A positive slope means the line goes up to the right; a negative slope means it goes down.

Property

In a linear relationship, the rate of change between the two variables is constant.
For any two points, the ratio of the change in the vertical variable to the change in the horizontal variable is always the same.
This constant rate of change is the slope of the line.
A positive slope indicates an increasing relationship, while a negative slope indicates a decreasing one.

Examples