Grade 10

Functions & Graphs

Understand the four basic function families — how they look, what affects their shape, and how to find their equations from a graph.

Linear Functions y = mx + c

A linear function gives a straight line. The two numbers that define it are:

Theorem

y = mx + c

$m$ — the gradient (slope). Positive = goes up left-to-right; negative = goes down.
$c$ — the y-intercept. Where the line crosses the y-axis.

Worked Example

Sketch $y = 2x - 3$ and find its intercepts.

To find the gradient between two points

(x_1,y_1)

and

(x_2,y_2)

use

m = \dfrac{y_2 - y_1}{x_2 - x_1}

Worked Example

A straight line passes through $(2,\ 1)$ and $(5,\ 7)$ . Find its equation.

Worked Example

A line has x-intercept 4 and y-intercept $-2$ . Find its gradient and equation.

Tip

Parallel lines have the same gradient. Perpendicular lines have gradients that multiply to

-1

— so if one line has gradient

m

, the perpendicular has gradient

-\tfrac{1}{m}