Understanding Slope

Learn what slope means, how to calculate it, and why it's essential for describing linear relationships.

Intermediate25 minLesson
Printable worksheet

Definition

Slope measures how steep a line is. It tells us how much a line rises or falls as we move from left to right.
Slope is calculated as:
slope=riserun=vertical changehorizontal change=y2−y1x2−x1\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\text{vertical change}}{\text{horizontal change}} = \frac{y_2 - y_1}{x_2 - x_1}
We often use the letter mm to represent slope:
m=ΔyΔxm = \frac{\Delta y}{\Delta x}
where Δ\Delta (delta) means "change in."

Try it now

What does slope measure?

Worked Examples

Find the slope of the line passing through (2,3)(2, 3) and (5,9)(5, 9).
1

Identify the coordinates

(x1,y1)=(2,3)(x_1, y_1) = (2, 3) and (x2,y2)=(5,9)(x_2, y_2) = (5, 9) → Points labeled
2

Find the rise (change in y)

y2−y1=9−3=6y_2 - y_1 = 9 - 3 = 6 → Rise = 6
3

Find the run (change in x)

x2−x1=5−2=3x_2 - x_1 = 5 - 2 = 3 → Run = 3
4

Calculate slope

m=riserun=63=2m = \frac{\text{rise}}{\text{run}} = \frac{6}{3} = 2 → m=2m = 2

Common Mistakes

Mixing up rise and run in the formula

Why it's wrong: Students sometimes calculate runrise\frac{\text{run}}{\text{rise}} instead of riserun\frac{\text{rise}}{\text{run}}.

Correct: Remember: slope = riserun\frac{\text{rise}}{\text{run}} = y2−y1x2−x1\frac{y_2 - y_1}{x_2 - x_1}. The y-values go on top!

Subtracting coordinates in different orders

Why it's wrong: Using (y2−y1)(y_2 - y_1) but (x1−x2)(x_1 - x_2) gives the wrong sign.

Correct: Always subtract in the same order: if you do y2−y1y_2 - y_1, you must also do x2−x1x_2 - x_1.

Saying vertical lines have slope = 0

Why it's wrong: Confusing "no slope" with "zero slope."

Correct: Horizontal lines have slope = 0. Vertical lines have undefined slope (cannot divide by zero).

Forgetting the negative sign

Why it's wrong: Not noticing when the line goes downward.

Correct: If the line falls from left to right, the slope is negative. Always check your sign!

Interactive Visual

Linear Function Explorer

y = x
Slope (m)1
Y-Intercept (b)0
b
run
rise
xy
-2-2
-1-1
00
11
22

Function graph

024680510xy(2, 3)(5, 9)run = 3rise = 6

Tap or hover over the graph to read off values.

y = 2x - 1
  • Slope of (2, 3) – (5, 9): m = 6/3 = 2

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

Tap or hover over the graph to read off values.

Try these examples:

y = 2x + 1

m=2, b=1

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy
What does slope measure?

Why It Matters

Slope is everywhere in real life:
  • Roads: A road with a 6% grade rises 6 feet for every 100 feet of horizontal distance
  • Ramps: Wheelchair ramps must have a slope no steeper than 112\frac{1}{12}
  • Economics: The slope of a cost graph tells you the price per item
  • Science: Velocity is the slope of a position-time graph
Understanding slope helps you interpret graphs, predict trends, and solve real-world problems!

Real World Applications

Road Grades and Highways

Road signs show grade as a percentage, which is slope expressed as rise per 100 units of run.

Example:

A 7% grade means the road rises 7 meters for every 100 meters of horizontal distance: slope = 7100=0.07\frac{7}{100} = 0.07

1Try It Yourself

A mountain road has a 12% grade. A cyclist travels 500 meters horizontally.

How many meters did the cyclist climb in elevation?

Step 1: Write the mathematical expression

12% means rise = 0.12 times run:

Phone Data Usage

Your phone plan charges a constant rate per gigabyte. The slope of your bill graph is the cost per GB.

Example:

If your bill goes from 20 euros to 35 euros when you use 3 extra GB, the slope is 35−203=5\frac{35-20}{3} = 5 euros per GB.

2Try It Yourself

At 2 GB usage, your bill is 15 euros. At 6 GB, it's 35 euros.

What is the cost per gigabyte?

Step 1: Write the mathematical expression

Find slope: change in costchange in data\frac{\text{change in cost}}{\text{change in data}}

Filling a Pool

When filling a pool at a constant rate, the slope of the water level graph tells you liters per minute.

Example:

If the water level rises from 50 cm to 90 cm in 20 minutes, the slope is 4020=2\frac{40}{20} = 2 cm per minute.

3Try It Yourself

A pool starts at 30 cm water level. After 45 minutes, it's at 120 cm.

How fast is the pool filling (cm per minute)?

Step 1: Write the mathematical expression

Slope = rise in water leveltime\frac{\text{rise in water level}}{\text{time}}

Key Takeaways

  • 1Slope measures steepness: m=riserun=y2−y1x2−x1m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}
  • 2Positive slope: line goes up from left to right
  • 3Negative slope: line goes down from left to right
  • 4Zero slope: horizontal line (no rise)
  • 5Undefined slope: vertical line (no run, division by zero)
  • 6Slope represents rate of change in real-world situations

Frequently Asked Questions

No! You can pick either point as point 1. Just be consistent: subtract in the same order for both x and y.
No! You can pick either point as point 1. Just be consistent: subtract in the same order for both x and y.
That's perfectly fine! A slope of 23\frac{2}{3} means "rise 2, run 3." You can leave it as a fraction or convert to a decimal (about 0.67).
It depends on context! A steeper hill is harder to climb. A steeper savings rate means you save faster. Interpret slope based on what the graph represents.

Glossary

Slope
A measure of how steep a line is, calculated as rise over run
Rise
The vertical change between two points (change in y)
Run
The horizontal change between two points (change in x)
Rate of change
How quickly one quantity changes relative to another; slope is a rate of change
Positive slope
A line that goes upward from left to right
Negative slope
A line that goes downward from left to right
Zero slope
A horizontal line where the rise equals zero
Undefined slope
A vertical line where the run equals zero (cannot divide by zero)

Formula Card

Slope (rise over run)

m=riserunm = \frac{\text{rise}}{\text{run}}

Slope equals vertical change divided by horizontal change

Slope (coordinate form)

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

Calculate slope using two points on the line

Slope (delta notation)

m=ΔyΔxm = \frac{\Delta y}{\Delta x}

Delta means change in; this is the same as rise over run

Today's challengeOne problem, three tries

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