Core Concept
What is Slope?
Slope (m) is the rate of change of y with respect to x. It measures rise (vertical change) over run (horizontal change).
m = rise/run = (y₂ − y₁) / (x₂ − x₁) = Δy/Δx
Types of Slope
What Different Slopes Look Like
Positive slope (m > 0)
Line rises left to right ↗
Negative slope (m < 0)
Line falls left to right ↘
Zero slope (m = 0)
Horizontal line →
Undefined slope
Vertical line ↑ (division by zero)
Example
Calculating Slope from Two Points
Find the slope through (1, 3) and (4, 9).
1. Label points: (x₁, y₁) = (1, 3) and (x₂, y₂) = (4, 9)
2. m = (y₂ − y₁) / (x₂ − x₁)
3. = (9 − 3) / (4 − 1)
4. = 6 / 3 = 2
Interpretation: for every 1 unit you move right, y increases by 2.
Watch Out
Common Mistakes
- Mixing up which is x₁ and y₁ — be consistent
- Writing rise/run upside down (run/rise)
- Confusing a zero slope (horizontal) with an undefined slope (vertical)
- m = (y change) / (x change). Vertical = undefined. Horizontal = 0.
Your Turn
Try It Yourself
Q1. Find the slope through (2, 5) and (6, 13).
Show Answer
m = (13−5)/(6−2) = 8/4 = 2
Q2. What is the slope of a horizontal line? What about a vertical line?
Show Answer
Horizontal: m = 0. Vertical: slope is undefined.
Key Takeaways
1-Minute Summary
- m = (y₂ − y₁) / (x₂ − x₁) = rise/run
- Positive → rises; Negative → falls; Zero → horizontal; Undefined → vertical
- Slope is a rate of change — "y increases/decreases by m for each unit of x"