Algebra
Linear functions
Slope
19 practice questions
2 video lessons
Theory + worked examples
Theory
Slope \(m\) measures steepness as rise over run:
\[m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}.\]
- Positive: rises left to right.
- Negative: falls.
- Zero: horizontal; undefined: vertical.
Subtract coordinates in the same order top and bottom.
Slope is the rise over the run.
The slope formula.
The slope formula:
\[m=\dfrac{y_2-y_1}{x_2-x_1}\]
Slope is the constant rate of change of a line.
How to find slope
- Pick two points \((x_1,y_1)\) and \((x_2,y_2)\).
- Compute the rise \(y_2-y_1\).
- Compute the run \(x_2-x_1\).
- Divide rise by run.
Example 1 — From two points
Find the slope through \((1,2)\) and \((3,8)\).
Solution
Use \(\dfrac{y_2-y_1}{x_2-x_1}\).
| \(m\) | \(=\) | \(\dfrac{8-2}{3-1}\) |
| \(=\) | \(\dfrac{6}{2}=3\) |
Example 2 — Negative slope
Find the slope through \((0,5)\) and \((2,1)\).
Solution
Compute the rise over run.
| \(m\) | \(=\) | \(\dfrac{1-5}{2-0}\) |
| \(=\) | \(-2\) |
Example 3 — Zero slope
What is the slope through \((1,4)\) and \((5,4)\)?
Solution
No change in \(y\).
| \(m\) | \(=\) | \(\dfrac{0}{4}=0\) |
Example 4 — Interpret slope
A line rises \(3\) for every \(1\) across. What is the slope?
Solution
Rise over run.
| \(m\) | \(=\) | \(\dfrac{3}{1}=3\) |
Common pitfalls
Keep the order consistent in the numerator and denominator.
Rise over run, not run over rise.
Zero slope \(\neq\) undefined slope.
Frequently asked questions
What is slope?
The steepness of a line, rise over run.
What is the slope formula?
\(m=\dfrac{y_2-y_1}{x_2-x_1}\).
What does a negative slope mean?
The line falls from left to right.
What is the slope of a horizontal line?
Zero.
More in Linear functions