Linear functions and straight-line graphs

A linear function has a constant rate of change. Written as $y = mx + c$ (the SAT writes $y = mx + b$):

• $m$, the gradient or slope, tells you how much $y$ changes when $x$ increases by $1$.
• $c$, the $y$-intercept, is the value of $y$ when $x = 0$.

Between points $(x_1, y_1)$ and $(x_2, y_2)$: $m = \frac{y_2 - y_1}{x_2 - x_1}$

1. Find the gradient $m$.
2. Substitute one known point into $y = mx + c$ to find $c$.

For a line with gradient $3$ through $(2, 7)$: $7 = 3(2) + c$, so $c = 1$ and the line is $y = 3x + 1$.

In context, the gradient is a rate (e.g. cost per gigabyte) and the intercept is a starting value (e.g. a fixed monthly fee).

Parallel lines have the same gradient. Perpendicular lines have gradients that are negative reciprocals: if one gradient is $\frac{2}{3}$, the other is $-\frac{3}{2}$.

Rearrange equations into $y = mx + c$ form before reading off the gradient: $2x + 3y = 6$ becomes $y = -\frac{2}{3}x + 2$.