How coefficients change a graph | Year 8

01 / LEARN

The idea, step by step

The two coefficients have different roles

In $y = m x + b$ , b is the output when x = 0. Changing b adds the same amount to every output. The coefficient m is the output change when x increases by 1.

Use a table to see these effects. A negative m means outputs decrease as x increases; it does not mean every output is negative.

Compare one change at a time

The rules $y = x + 1$ and $y = 3 x + 1$ contain the same point at x = 0 but rise at different rates per input step. The rules $y = 2 x + 1$ and $y = 2 x - 2$ rise at the same rate and remain 3 units apart.

Use consistent axes when comparing pictures. For negative inputs, a larger positive multiplier can produce a smaller output, so avoid a claim about every output based only on the coefficient size.

A common trap

Assuming a larger multiplier always means a larger output.

Instead: Compare specified inputs; negative inputs can reverse the comparison.

02 / WORKED EXAMPLES

See the thinking

Example 1

Change the constant

Compare $y = x$ and $y = x + 2$ .

At x = -1, 0 and 1, the first outputs are -1, 0 and 1.
The second outputs are 1, 2 and 3: always 2 more.
Their rise per input step is unchanged.

The second graph shifts up 2 units.

Example 2

Change direction

Compare $y = 2 x + 1$ and $y = - x + 1$ .

At x = 0 both give 1.
Moving to x = 1 gives 3 for the first and 0 for the second.
This separates starting value from direction.

They share

(0, 1)

; one rises by 2 and the other falls by 1 per input step.

Check your understanding

Describe the change from $y = 3 x + 2$ to $y = 3 x - 4$ .

Show answer

Every output decreases by 6.

Only the constant changes, from 2 to -4.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

What is the output at x = 0 for $y = 4 x - 3$ ?
2 marks
For $y = 5 x + 2$ , what happens to y when x increases by 1?
2 marks
Describe the change from $y = 2 x - 1$ to $y = 2 x + 3$ .
2 marks
Compare $y = x + 1$ and $y = 3 x + 1$ at x = 2 and x = -2.
3 marks
Which of $y = 2 x + 5$ and $y = - 2 x + 5$ decreases as x increases?
2 marks
Explain why $y = 3 x + 1$ and $y = 3 x - 2$ never meet.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

What is the output at x = 0 for $y = 4 x - 3$ ?
2 marks
Substitute zero.
The result is $0 - 3$ .
-3.
For $y = 5 x + 2$ , what happens to y when x increases by 1?
2 marks
The coefficient is 5.
For example outputs 2 and 7 occur at x = 0 and 1.
It increases by 5.
Describe the change from $y = 2 x - 1$ to $y = 2 x + 3$ .
2 marks
The multiplier is unchanged.
The constant increases by 4.
An upward shift of 4.
Compare $y = x + 1$ and $y = 3 x + 1$ at x = 2 and x = -2.
3 marks
Substitute into each rule independently.
The larger coefficient does not always give a larger output.
At 2: 3 and 7. At -2: -1 and -5.
Which of $y = 2 x + 5$ and $y = - 2 x + 5$ decreases as x increases?
2 marks
The multiplier is negative.
Each extra input unit reduces y by 2.
$y = - 2 x + 5$ .
Explain why $y = 3 x + 1$ and $y = 3 x - 2$ never meet.
3 marks
The multipliers are equal.
Their different constants prevent equality.
Their outputs remain 3 apart for the same input.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics (2026)

Multi-step equations; brackets and variables on both sides; equations with rational solutions; substitution checks; one-variable inequalities and number-line solutions; tables and straight-line graphs; horizontal/vertical lines; how coefficients change a graph; graphical solutions and intersection checks; real-world linear models with digital exploration.

Official curriculum source ↗

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