Linear Functions & Their Graphs
“A linear function is a straight line; the slope is 'how steep'.”
The formula
y = ax + bHow to read it: a straight line with slope a and y-intercept b
- a
- — the slope — how much y rises when x increases by 1
- b
- — the y-intercept — the value of y when x = 0 (the starting height)
- x
- — the input value
The hook
y=ax+b is just a straight line — the slope a sets 'how steep' it is, and b sets 'where it starts'.
In plain words
A rule where you feed in x and out comes y = 'a times x plus b'. Graph it and you always get a straight line.
The intuition
a is the steepness of a staircase — every time x rises by 1, y climbs by a. b is the starting height (y when x=0). Know those two and the line is fully fixed.
How it's built
a is the slope = (change in y)/(change in x). b is the y-intercept, the height where the line crosses the y-axis. If a is positive the line heads up to the right, if negative it heads down.
Example
y=2x−1. At x=0,1,2, y is −1,1,3. Each time x rises by 1, y rises by 2 — the slope is exactly 2.
Common mistake
It's easy to confuse the slope a with the intercept b. In y=3x−2, 3 is the slope and −2 is the starting height — not 3.
Where it's used
Phone plans (base fee + usage), distance at constant speed, temperature change — the basic model for any relationship that 'changes at a steady rate'.
Where it came from
The coordinate plane that links equations to graphs was created by Descartes in the 1600s — the moment an equation could finally become a 'picture'.
Prerequisites
Quick check
What is the slope of the linear function y = 3x − 2?
- 3✓
- −2
- 2
- −3
Practice
For y = 2x + 1, what is y when x = 3?
Answer: 7
- y = 2·3 + 1
- = 7
Takeaway: Plug in x and out comes a point y on the line.
For y = −2x + 5, what is y when x = 1?
Answer: 3
- y = −2·1 + 5
- = 3
Takeaway: A negative slope means y falls as x rises.
What is the slope of the line through (1, 2) and (3, 8)?
Answer: 3
- slope = (change in y)/(change in x) = (8 − 2)/(3 − 1)
- = 6/2 = 3
Takeaway: Slope is the rise between two points divided by the run.