Lesson 4: Slope

New Concept

Slope measures a line's steepness, but it's more than that—it's a rate of change. We will calculate slope using $m = \frac{\Delta y}{\Delta x}$ and apply it to understand real-world rates like speed, growth, and accumulation.

What’s next

Now, let's put this into action. You'll tackle interactive examples and practice cards to master calculating and interpreting slope in different contexts.

Property

A rate is a type of ratio that compares two quantities with different units.

A rate of change is a special kind of ratio that compares the change in two quantities or variables.

In mathematics, we use the symbol $\Delta$ (delta) for change in. Thus, a rate of change can be expressed as the ratio of the change in one variable to the change in another:

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

1. The slope of a line measures the rate of change of $y$ with respect to $x$.
2. The units of $\Delta y$ and $\Delta x$ can help us interpret the slope as a rate.

Examples

• If a graph plots distance (in miles) vs. time (in hours), a slope of 65 means the speed is 65 miles per hour.
• A graph shows a phone's battery percentage vs. time in hours. A slope of $-10$ means the battery is draining at a rate of 10 percent per hour.
• A company's cost to produce widgets is graphed with cost (in dollars) on the y-axis and number of widgets on the x-axis. A slope of 1.5 means each additional widget costs 1.50 dollars to produce.

Explanation

The slope's number gets its real-world meaning from the units on the axes. By combining the y-axis unit 'per' the x-axis unit, you can explain exactly what the rate of change signifies, like 'cost per ticket' or 'feet per second'.

Property

The slope of a line is constant: no matter which two points you pick to compute the slope, you will always get the same value.

Because $m$ is constant for a given line, we can use the formula $m = \frac{\Delta y}{\Delta x}$ to find $\Delta y$ when we know $\Delta x$, or to find $\Delta x$ when we know $\Delta y$.

We can also tell whether a collection of data points lies on a straight line by computing slopes between them.