Lesson 41: Finding Rates of Change and Slope

New Concept

Algebra explores relationships. A key tool is the rate of change, a ratio comparing how two quantities change. Visually, this is represented by a line's slope.

What’s next

This is your foundation. Next, you'll master calculating rates and slope from graphs, tables, and points, bringing these abstract concepts to life with practical examples.

Property

A rate of change is a ratio that compares the change in one quantity with the change in another.

Examples

• A car travels from mile marker 50 to mile marker 200 in 3 hours: $\frac{200 - 50}{3} = \frac{150}{3} = 50$ miles per hour.
• The cost to rent a scooter changes from 30 dollars to 75 dollars over 3 hours: $\frac{75 - 30}{5 - 2} = \frac{45}{3} = 15$ dollars per hour.

Explanation

This just means comparing how one thing changes with another, like miles per hour or dollars per day. Think of it as the 'story' behind the numbers. If your allowance goes up 5 dollars every 2 weeks, the rate of change tells you exactly how fast your fortune is growing. It is the real-world speed of change!

Property

The slope of a line is a rate of change. It is equal to the ratio of the vertical change (rise) to the horizontal change (run).

Examples

• A line rises 8 units for every 2 units it runs to the right: $\operatorname{slope} = \frac{8}{2} = 4$.
• A line falls 9 units for every 3 units it runs to the right: $\operatorname{slope} = \frac{-9}{3} = -3$.
• A line passes through (1, 2) and (5, 10): $\operatorname{slope} = \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2$.

Explanation

Slope is just a fancy word for how steep a line is on a graph. It is the ultimate measure of its tilt! We call the vertical change the 'rise' (like climbing a ladder) and the horizontal change the 'run' (like running across a field). A big slope means a super steep hill, while a small slope is a gentle walk.

Property

The slope of a horizontal line is 0. The slope of a vertical line is undefined.

Examples

• The slope of a horizontal line passing through (1, 3) and (7, 3) is $\frac{3 - 3}{7 - 1} = \frac{0}{6} = 0$.
• The slope of a vertical line passing through (5, 2) and (5, 9) is $\frac{9 - 2}{5 - 5} = \frac{7}{0}$, which is undefined.

Explanation

Imagine you are on a skateboard. A horizontal line is perfectly flat ground—zero slope means zero effort required to just stand still! A vertical line is like trying to skate directly up a wall. It is so steep it is impossible, which is why we say its slope is 'undefined.' You cannot put a number on that level of steepness!