Slope measures how much y changes for each unit of x. It is rise over run.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
From two points
For (x₁, y₁) and (x₂, y₂), slope is (y₂ − y₁) ÷ (x₂ − x₁).
Keep the order consistent
Whichever point you call first, use it first in both subtractions. Mixing them flips the sign.
What the sign means
Positive slope rises left to right, negative falls. Zero slope is horizontal, and a vertical line has undefined slope.
Slope has units
In a cost graph the slope is dollars per item. Naming the units turns a number into a sentence.
Linear means constant
A relationship is linear exactly when the rate of change never varies. A table with equal x-steps and equal y-steps is linear.
Steepness and rate are the same number
Slope is rise over run geometrically and change in output per unit input in context. The integrated course puts both readings side by side because they are the same quantity seen from two directions.
Slope from two points
(y₂ − y₁)/(x₂ − x₁). Use the same point first in both subtractions; mixing the order flips the sign. Either point may go first provided you are consistent.
The units name what it measures
Output units per input unit — miles per hour, dollars per item. Stating the units is usually the most informative thing you can say about a slope in a modelling context.
Zero and undefined
A horizontal line has slope zero; a vertical line has undefined slope because the run is zero. These are genuinely different cases, and calling a vertical line infinitely steep is imprecise rather than informal.
Step 2: Try It Yourself
Tap and try it out.
Both triangles give the same slope, 2, even though one is bigger. They are the same shape, so their sides are in the same ratio.
Step 3: Watch an Example
One step at a time.
Watch Marcus Find a Rate
A taxi costs $8 for 2 miles and $20 for 8 miles.
- Step 1
The two points are (2, 8) and (8, 20).
Step 4: Your Turn
Practice makes it stick.
The Two Points
Problem 1 of 2
Through (1, 3) and (4, 12). What is the slope?
The Taxi
Problem 2 of 2
$8 for 2 miles and $20 for 8 miles. Cost per mile, in dollars?
dollars
Rise Over Run
1 of 8
Through (0, 0) and (2, 6). Slope?
2 of 8
Through (1, 5) and (3, 5). Slope?
3 of 8
Through (2, 8) and (5, 2). Slope?
4 of 8
Through (−1, 2) and (3, 10). Slope?
5 of 8
A horizontal line. What is its slope?
6 of 8
Through (4, 1) and (4, 9). Is the slope defined? 1 yes, 0 no.
7 of 8
Sort each slope by the direction of the line.
Tap something to move it.
- Empty
- Empty
8 of 8
Through (0, 7) and (4, 3). Slope?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Through (2, 4) and (6, 16). What is the slope?
Question 2 of 2
What makes a relationship linear?
What You Learned
- Slope is rise over run, or (y₂ − y₁) ÷ (x₂ − x₁).
- Its sign gives the direction; zero is horizontal and vertical is undefined.
- A relationship is linear exactly when the rate of change is constant.