Gradient: up over across
The gradient measures steepness: how far the line goes up for every 1 across.
gradient = change in y ÷ change in x
- A(1, 2) and B(5, 10)
- change in y: 10 − 2 = 8
- change in x: 5 − 1 = 4
- gradient = 8 ÷ 4 = 2
A line sloping down from left to right has a negative gradient. Gradients can be fractions: from (0, 1) to (4, 4) is 3 up for 4 across, a gradient of 3/4. Subtract in the same order top and bottom, and always put the change in y on top.
Reading m and c
In y = mx + c, m is the gradient and c is the y-intercept: the line crosses the y-axis at (0, c).
These are y = 2x − 3 (steep, up, crossing at −3) and y = −x/2 + 4 (gentle, down, crossing at 4). When the equation is in another form, get y on its own first:
- 2y − 4x = 6
- add 4x: 2y = 4x + 6
- divide every term by 2: y = 2x + 3
- gradient 2, y-intercept 3
Is a point on the line?
A point lies on a line when its coordinates make the equation true. Is (3, 5) on y = 2x − 1? 2 × 3 − 1 = 5, so yes. Is (−2, 3)? 2 × (−2) − 1 = −5, not 3, so no.
The slip to watch: in y = 5 − 2x the gradient is −2, not 5. The sign in front of the x term belongs to the gradient.
Drive the line with m and c
Set m to 2 and c to −3 and check the line crosses the y-axis at −3 and rises 2 for every 1 across. Now set m to −1/2 and c to 4. Then keep c at 4 and try m = 0: which kind of line do you get, and what is its equation?