Functions

Linear functions

A linear function has the form $y = mx + b$ . Its graph is a straight line: $m$ is the slope — how much $y$ changes when $x$ grows by 1 — and $b$ is where the line crosses the $y$ -axis.

What is a linear function?

y = mx + b

Slope $m$
Positive: the line rises from left to right. Negative: it falls. Zero: it’s horizontal. Bigger $|m|$ means steeper.
Intercept $b$
The value of $y$ when $x = 0$ — the point $(0, b)$ on the $y$ -axis.
Constant rate
Every step of $1$ in $x$ changes $y$ by the same amount, $m$ . That’s what makes it a straight line.

Explore the graph

Change the slope and the intercept and watch the line respond. Then try the challenge.

Worked example

A line through two points

Problem: Find the equation of the line through $(1, 3)$ and $(4, 9)$ .

Slope = change in $y$ divided by change in $x$ .
$m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$
Substitute one point into $y = 2x + b$ .
$3 = 2 \cdot 1 + b \;\Rightarrow\; b = 1$
Check with the other point.
$2 \cdot 4 + 1 = 9 \;\checkmark$

Result: $y = 2x + 1$

Common mistakes

Mistake 1 · Mixing the order in the slope formula: $\frac{9 - 3}{1 - 4}$ .
Why it happens
Top and bottom must subtract the points in the same order.
What to do instead
$\frac{y_2 - y_1}{x_2 - x_1}$ — same point first in both.
Mistake 2 · Confusing $b$ with where the line crosses the $x$ -axis.
Why it happens
$b$ is the $y$ -intercept: the value at $x = 0$ .
What to do instead
For the $x$ -intercept, set $y = 0$ and solve.
Mistake 3 · Thinking a bigger $b$ makes the line steeper.
Why it happens
$b$ only shifts the line up or down.
What to do instead
Steepness comes from $m$ .

Practice

1. What are the slope and $y$ -intercept of $y = -3x + 4$ ?
Hint
Compare with $y = mx + b$ .
Show answer
Slope $-3$ , intercept $4$ .
2. Find the line through $(0, 2)$ and $(2, 8)$ .
Hint
The first point already gives you $b$ .
Show answer
$m = \frac{8 - 2}{2 - 0} = 3$ , so $y = 3x + 2$ .
3. Is the point $(3, 7)$ on the line $y = 2x + 1$ ?
Hint
Substitute $x = 3$ .
Show answer
Yes: $2 \cdot 3 + 1 = 7$ .
4. Where does $y = 2x - 6$ cross the $x$ -axis?
Hint
Set $y = 0$ .
Show answer
$0 = 2x - 6$ , so $x = 3$ : the point $(3, 0)$ .

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Linear functions

What is a linear function?

Slope $m$

Intercept $b$

Constant rate