What does gradient tell you?
Gradient describes how steep a straight line is. It compares the change up or down with the change across. A positive gradient rises as you move right; a negative gradient falls.
If you know two points on a line, subtract their y-coordinates and divide by the difference between their x-coordinates. Keep the subtraction in the same order for both.
Follow a worked example
Suppose a line passes through (2, 3) and (6, 11). We want its gradient.
Find the vertical change: 11 − 3 = 8.
Find the horizontal change: 6 − 2 = 4.
Divide: 8 ÷ 4 = 2. The gradient is 2.
Notice that changing the order of the points gives the same answer: (3 − 11) ÷ (2 − 6) = −8 ÷ −4 = 2. What matters is keeping the order consistent.
Try it yourself
Give each question a go before opening its answer. Paper and pencil are enough.
1. What is the gradient through (1, 2) and (4, 8)?
Show answer and method
Change in y: 8 − 2 = 6. Change in x: 4 − 1 = 3. Gradient = 6 ÷ 3 = 2.
2. What is the gradient through (−2, 5) and (2, −3)?
Show answer and method
Change in y: −3 − 5 = −8. Change in x: 2 − (−2) = 4. Gradient = −8 ÷ 4 = −2. The negative sign means the line falls as you move right.
3. In y = 3x − 4, what are the gradient and y-intercept?
Show answer and method
The equation is in the form y = mx + c. Gradient m = 3 and y-intercept c = −4. The line crosses the y-axis at (0, −4).
Take the method with you
Write “change in y ÷ change in x” beside your next graph question. If your answer has the wrong sign, check the subtraction order before you start again.