How coefficients change a graph | Year 8

01 / LEARN

Understand the idea.

The two coefficients have different roles

In $y = m x + b$ , the constant $b$ is the output at $x = 0$ , so the graph meets the vertical axis at $(0, b)$ . Changing $b$ adds the same amount to every output.

The coefficient $m$ is the output change when $x$ increases by $1$ . Positive $m$ gives a rising line; negative $m$ gives a falling line. A falling line can still include positive outputs.

Compare one change at a time

Compare one coefficient at a time. The lines $y = x + 1$ and $y = 3 x + 1$ share their vertical-axis crossing but rise at different rates. The lines $y = 2 x + 1$ and $y = 2 x - 2$ have equal rates and stay parallel.

Use the same axis scales to compare steepness. A larger positive coefficient does not make every output larger: for negative inputs, multiplying by a larger positive number gives a more negative product.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Change the constant

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

Equal-scale Cartesian axes from negative 4 to 4 show the sage line y equals x and the gold line y equals x plus 2, each clipped at the grid boundary.

Worked solution

1
Compare outputs at the same input.
$(x + 2) - x = 2$
The second rule gives an output exactly $2$ greater for every $x$ , not just selected examples.
2
Check representative coordinates.
$\begin{aligned} y = x : & - 1, 0, 1 \\ y = x + 2 : & 1, 2, 3 \end{aligned}$
These rows use inputs $- 1$ , $0$ , $1$ in the same order and show the same vertical difference.
3
Describe the graph change.
$m_{1} = m_{2} = 1$
The rate is unchanged, so the second line is the first line shifted up $2$ units.

Answer

The second graph shifts up 2 units.

EXAMPLE 02

Change direction

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

Worked solution

1
Find the common vertical-axis crossing.
$\begin{aligned} 2 (0) + 1 & = 1 \\ - (0) + 1 & = 1 \end{aligned}$
At $x = 0$ , both rules give the point $(0, 1)$ .
2
Read the rates from the coefficients.
$m_{1} = 2, m_{2} = - 1$
For each unit increase in input, the first output rises by two and the second falls by one.
3
Check one input step.
$\begin{aligned} y_{1} (1) & = 2 (1) + 1 = 3 \\ y_{2} (1) & = - 1 + 1 = 0 \end{aligned}$
Starting from output $1$ , the outputs change by $+ 2$ and $- 1$ respectively.

Answer

They share $(0, 1)$ ; one rises by 2 and the other falls by 1 per input step.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Every output decreases by 6.

The coefficient stays $3$ , while the constant changes by $- 4 - 2 = - 6$ . Every output decreases by $6$ , so the graph shifts down $6$ units with its rate unchanged.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 $x = 0, 1$ .
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 ↗

In this lesson

Understand the idea.

The two coefficients have different roles

Compare one change at a time

See the method in action.

Change the constant

Compare outputs at the same input.

Check representative coordinates.

Describe the graph change.

Change direction

Find the common vertical-axis crossing.

Read the rates from the coefficients.

Check one input step.

Try it yourself.

Check your understanding

Keep practising