Year 9: Calculating the gradient of a line

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1. What is the gradient?

The gradient (or slope) of a straight line shows how steep the line is. It tells us how much the y-value changes for each unit increase in the x-value.

The gradient is usually represented by the letter m.

2. Gradient formula

If you know two points on a line: (x₁, y₁) and (x₂, y₂), then the gradient m is calculated as:

m = (y₂ − y₁) ÷ (x₂ − x₁)

3. Positive and negative gradients

• Positive gradient: The line rises from left to right.
• Negative gradient: The line falls from left to right.
• Zero gradient: The line is horizontal.
• Undefined gradient: The line is vertical (x₁ = x₂).

4. Example 1 – Find the gradient

Find the gradient of the line passing through the points (2, 3) and (6, 11).

Using the formula:

m = (11 − 3) ÷ (6 − 2) = 8 ÷ 4 = 2

5. Example 2 – Negative gradient

Find the gradient of the line passing through the points (5, 10) and (9, 2).

m = (2 − 10) ÷ (9 − 5) = −8 ÷ 4 = −2

6. Special cases

• Points (3, 5) and (7, 5): m = (5 − 5) ÷ (7 − 3) = 0 → horizontal line
• Points (4, 1) and (4, 6): m = (6 − 1) ÷ (4 − 4) = 5 ÷ 0 → undefined

7. Using gradient to draw lines

If a line has a gradient of m and passes through a point (x₁, y₁), you can use the gradient to find other points by moving:

• Up or down depending on the numerator (change in y)
• Left or right depending on the denominator (change in x)

Example: Gradient = 2 means rise 2 units for every 1 unit right.

8. Practice questions

1. Find the gradient of the line passing through (1, 4) and (5, 12).
2. Find the gradient between the points (−2, 6) and (4, −6).
3. What is the gradient of the line through (7, 3) and (7, −1)?
4. Is the gradient of a line through (−1, −2) and (2, −2) positive, negative, zero, or undefined?

9. Solutions

1. m = (12 − 4) ÷ (5 − 1) = 8 ÷ 4 = 2
2. m = (−6 − 6) ÷ (4 + 2) = −12 ÷ 6 = −2
3. m = (−1 − 3) ÷ (7 − 7) = −4 ÷ 0 → undefined
4. m = (−2 − (−2)) ÷ (2 + 1) = 0 ÷ 3 = 0 → zero gradient