y = mx + c
- The equation of a straight line can be written y = mx + c, where m is the gradient and c is the y-intercept (where the line crosses the y-axis).
- y = 3x − 2 has gradient 3 and crosses the y-axis at (0, −2).
- Rearrange first if needed: 2y = 6x + 4 becomes y = 3x + 2.
Gradient
- Gradient = change in y ÷ change in x. Through (1, 2) and (3, 8): (8 − 2) ÷ (3 − 1) = 3.
- A positive gradient slopes up to the right; a negative gradient slopes down.
- Parallel lines have the same gradient.
- (Higher) Perpendicular lines have gradients that multiply to −1: the perpendicular to a line with gradient 2 has gradient −1/2.
Special lines and points
- y = a is a horizontal line; x = a is a vertical line.
- The x-intercept is where y = 0: y = 2x − 6 crosses the x-axis at (3, 0).
- The midpoint of (2, 4) and (6, 10) is ((2 + 6)/2, (4 + 10)/2) = (4, 7).
Finding the equation of a line
- Find the gradient, then substitute a point to find c: through (1, 5) and (3, 11), m = 3, and 5 = 3 × 1 + c gives c = 2, so y = 3x + 2.
- A point lies on a line if its coordinates make the equation true: (2, 7) is on y = 3x + 1 because 3 × 2 + 1 = 7.
Key terms
- Gradient
- The steepness of a line: change in y ÷ change in x.
- y-intercept
- The point where a line crosses the y-axis.
- x-intercept
- The point where a line crosses the x-axis.
- Parallel
- Lines with the same gradient, which never meet.
- Perpendicular
- At right angles; gradients multiply to −1.
- Midpoint
- The point exactly halfway between two points.