Slope Review & Rate of Change
Slope measures the steepness and direction of a line — the ratio of vertical change (rise) to horizontal change (run). It is the foundation of coordinate geometry.
Introduction
You first learned slope in algebra as “rise over run.” In geometry, slope connects coordinate algebra to parallel and perpendicular line relationships — the topics ahead.
Past Knowledge
Coordinate plane (1.2). Algebra 1 slope concepts.
Today's Goal
Calculate slope from two points and interpret slope as rate of change.
Future Success
Slopes of parallel lines (3.2.2) and perpendicular lines (3.2.3) depend on this formula.
Key Concepts
Slope Formula
Four Types of Slope
| Slope | Line Direction | Example |
|---|---|---|
| Positive | Rises left→right ↗ | |
| Negative | Falls left→right ↘ | |
| Zero | Horizontal → | |
| Undefined | Vertical ↑ | division by 0 |
Interactive Diagram — Desmos Geometry
Explore the rise and run of a line. Drag the points to change the slope and see how it affects the steepness.
Worked Examples
Slope from Two Points
Find the slope through and .
— the line rises 2 units for every 1 unit right.
Horizontal and Vertical Lines
Find the slope through and , then through and .
First pair: → horizontal line.
Second pair: → undefined (vertical line).
Rate of Change Application
A candle is 12 in. tall at 2 PM and 7 in. tall at 5 PM. What is the rate of change? What does the slope represent?
The candle burns at inches per hour. The negative slope means the height is decreasing.
Common Pitfalls
Swapping Rise and Run
Slope = (y first!). Writing gives you the reciprocal — a completely different slope.
Inconsistent Point Order
If you subtract on top, you must subtract on bottom. Mixing the order flips the sign.
Real-Life Applications
Road Grades
Road steepness is expressed as a percent grade — which is slope × 100. A “6% grade” means the road rises 6 feet for every 100 feet of horizontal distance, i.e., a slope of 0.06.
Practice Quiz
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