Testing parameter changes digitally | Year 9

01 / LEARN

Understand the idea.

Change one thing at a time

The parameters $m$ and $c$ control different features of $y = m x + c$ . Hold one fixed while changing the other, so a change in the graph can be traced to a single cause.

Predict first, then compare equations at matching inputs using a table or graphing tool. Record the axis scale and the parameter changed. Several examples help reveal a pattern; algebra is needed to justify a claim for every input.

Equal-unit Cartesian grid from negative six to eight on both axes, showing the labelled parallel lines y equals two x minus one in gold and y equals two x plus three in sage.

Only the intercept changes
x	y = 2x - 1	y = 2x + 3
-2	-5	-1
-1	-3	1
0	-1	3
1	1	5
2	3	7

Explain the pattern algebraically

Changing only $c$ adds the same amount to every output, producing a vertical shift and equal gradients. Changing only $m$ leaves the point $(0, c)$ fixed, because the $m x$ term vanishes there.

A larger gradient does not always give a higher graph. The sign of $x$ matters when comparing $m x$ . Use an output difference to explain when one rule is higher, or one counterexample to refute an "always" claim.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Predict an intercept change

Compare $y = 2 x - 1$ with $y = 2 x + 3$ . What changes?

Worked solution

1
Compare the gradients.
$m_{1} = 2, m_{2} = 2$
The coefficient of x is unchanged.
2
Subtract outputs at the same input.
$\begin{aligned} y_{2} - y_{1} & = (2 x + 3) - (2 x - 1) \\ = 4 \end{aligned}$
The x-terms cancel for every real input, not just the values in a table.
3
Describe the resulting shift.
$y_{2} = y_{1} + 4$
Every point moves up 4 units, so the two distinct lines remain parallel.

Answer

A vertical shift up 4 units; the gradient is unchanged.

EXAMPLE 02

Test a tempting claim

A student says $y = 3 x + 1$ is always above $y = x + 1$ . Test the claim.

Worked solution

1
Try a positive input.
$\begin{aligned} 3 (2) + 1 & = 7 \\ 2 + 1 & = 3 \end{aligned}$
At x equal to 2, the first rule is higher, but one supporting case does not prove "always".
2
Test a negative input.
$\begin{aligned} 3 (- 2) + 1 & = - 5 \\ - 2 + 1 & = - 1 \end{aligned}$
The negative input makes the larger x-coefficient produce a lower output.
3
Reject the universal claim.
$- 5 < - 1$
One counterexample is sufficient to disprove a statement about every input.
4
Explain the full comparison.
$\begin{aligned} (3 x + 1) - (x + 1) & = 2 x \\ x = 0 & \Rightarrow y = 1 \end{aligned}$
The first line is lower for negative x, higher for positive x, and equal at (0,1).

Answer

The claim is false; $x = - 2$ gives a counterexample.

03 / YOUR TURN

Try it yourself.

Check your understanding

If $y = - 2 x + 5$ changes to $y = - 2 x + 2$ , what happens to the graph?

Show answer & reasoning

It shifts down 3 units and remains parallel.

The output difference is $(- 2 x + 2) - (- 2 x + 5) = - 3$ for every $x$ . The line therefore moves down $3$ units and remains parallel because its gradient stays $- 2$ .

Keep practising

6 worksheet questions · 16 marks

Open worksheet Worked solutions

Read the questions online

Complete the outputs for $y = x + 2$ and $y = x - 1$ at $x = - 1, 0, 1$ .
3 marks
Predict the change from $y = 4 x - 2$ to $y = 4 x + 5$ without plotting.
2 marks
Where do $y = - x + 3$ , $y = 2 x + 3$ and $y = 5 x + 3$ all meet? Explain.
2 marks
For $y = m x + 2$ , which value $m = 1$ or $m = 3$ gives the larger output at $x = - 2$ ? Show both values.
3 marks
Design two distinct lines that meet at $(0, - 1)$ , then verify the common point.
3 marks
A graphing screen suggests $y = 0.5 x + 2$ and $y = 0.5 x + 2.1$ are the same line. Explain how to check.
3 marks

Read the worked solutions

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

Complete the outputs for $y = x + 2$ and $y = x - 1$ at $x = - 1, 0, 1$ .
3 marks
Substitute each input into both rules.
The first output is always 3 greater.
First rule: 1, 2, 3. Second rule: $- 2, - 1, 0$ .
Predict the change from $y = 4 x - 2$ to $y = 4 x + 5$ without plotting.
2 marks
The gradients are equal.
The intercept increases by $5 - (- 2) = 7$ .
A shift up 7 units; the lines are parallel.
Where do $y = - x + 3$ , $y = 2 x + 3$ and $y = 5 x + 3$ all meet? Explain.
2 marks
For $x = 0$ , the $m x$ term is zero in each rule.
All three rules then give $y = 3$ .
$(0, 3)$
For $y = m x + 2$ , which value $m = 1$ or $m = 3$ gives the larger output at $x = - 2$ ? Show both values.
3 marks
For $m = 1$ , $y = - 2 + 2 = 0$ .
For $m = 3$ , $y = - 6 + 2 = - 4$ .
$m = 1$ gives 0, which is larger than $- 4$ for $m = 3$ .
Design two distinct lines that meet at $(0, - 1)$ , then verify the common point.
3 marks
Choose different gradients with the same intercept $- 1$ .
Substituting $x = 0$ gives $y = - 1$ in both.
For example, $y = 2 x - 1$ and $y = - x - 1$ .
A graphing screen suggests $y = 0.5 x + 2$ and $y = 0.5 x + 2.1$ are the same line. Explain how to check.
3 marks
The same gradient means they are parallel.
The different intercepts show they are not the same line.
A large viewing window can hide a small separation.
Compare exact equations or substitute $x = 0$ : the outputs are 2 and 2.1, so the lines are distinct and parallel.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Gradient, midpoint and distance; algebraic and graphical straight-line relationships; interpreting models and testing parameters. Parallel and perpendicular lines are included as a natural school-course extension. All explanations and questions are original.

Official curriculum source ↗

In this lesson

Understand the idea.

Change one thing at a time

Explain the pattern algebraically

See the method in action.

Predict an intercept change

Compare the gradients.

Subtract outputs at the same input.

Describe the resulting shift.

Test a tempting claim

Try a positive input.

Test a negative input.

Reject the universal claim.

Explain the full comparison.

Try it yourself.

Check your understanding

Keep practising