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Key takeaways
- check_circleThe formula: Slope m = (y₂ − y₁) ÷ (x₂ − x₁), or rise ÷ run.
- check_circleExample: Between (2, 3) and (6, 11) the slope is 8 ÷ 4 = 2.
- check_circleLine equation: In y = mx + b, m is the slope and b is the y-intercept.
1. What is slope?
Slope measures how quickly a line rises or falls as you move from left to right. It is often represented by m and compares the change in y to the change in x.
Formula
m = (y₂ − y₁) ÷ (x₂ − x₁)
In words: slope = rise ÷ run.
2. How to find the slope between two points
Suppose the points are (2, 3) and (6, 11). Substitute the coordinates into the formula: (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2. The slope is 2.
3. Types of slope
- Positive: the line rises from left to right.
- Negative: the line falls from left to right.
- Zero: a horizontal line has slope 0.
- Undefined: a vertical line has no finite slope because the run is zero.
4. Slope and slope-intercept form
In the equation y = mx + b, m is the slope and b is the y-intercept. If m = 3, the line rises 3 units for every 1 unit it moves right.
5. Worked examples
| Points | Slope |
|---|---|
| (1, 2), (5, 10) | 2 |
| (2, 8), (6, 4) | −1 |
| (1, 5), (7, 5) | 0 |
6. Common mistakes
- Reversing the order of x and y differences.
- Using y₁ − y₂ while using x₂ − x₁, which changes the sign.
- Dividing by zero for a vertical line.
- Confusing slope with the y-intercept.
7. Frequently asked questions
What is the slope formula?expand_more
m = (y₂ − y₁) ÷ (x₂ − x₁).
How do you find slope between two points?expand_more
Subtract the y-coordinates, subtract the corresponding x-coordinates, and divide the first difference by the second.
What is the slope of a vertical line?expand_more
It is undefined because the change in x is zero and division by zero is not defined.