Parameter Variation and Graph Transformations — AC9M9A06

Changing parameters changes predictable graph features. Digital tools help test conjectures, but the goal is to connect graphical change to algebraic structure.

By the end of this lesson, you should be able to:

• vary one parameter at a time and predict graph changes
• connect parameters to intercepts, steepness, shifts or curvature
• use digital tools systematically rather than by trial-and-error
• generalise observed patterns and test counterexamples

Recall linear and quadratic graphs, gradient, intercepts, coordinates and substitution.

Control the experiment

Change one parameter while keeping others fixed so the cause of a graph change is visible.

Linear parameters have interpretable roles

In y=mx+c, m controls gradient and c controls the y-intercept.

Quadratic parameters change shape and position

In y=ax², a affects vertical stretch/reflection; in y=(x−h)²+k, h and k shift the turning point.

Generalisation requires testing

Test negative, zero, fractional and boundary values before stating a rule.

• Changing two parameters proves which caused the effect: Vary one at a time.
• Negative gradient means the graph is below the axis: It means y decreases as x increases.
• All effects are visible in any window: Poor graph windows can hide features.
• A digital pattern is proof: Support it with algebraic reasoning and tests.

1. Predict then graph y=x+1, y=x+4 and y=x−2.
2. Predict then graph y=x², y=3x² and y=−x².
3. Investigate y=(x−h)² for h=−2,0,3 and state a rule.
4. Test the rule with a new value.

Guided method: Name the mathematical structure first, show the calculation or representation, then verify with an estimate, inverse operation, second representation or digital check.

1. Describe c in y=4x+c.
2. Describe changing m from positive to negative in y=mx+2.
3. Compare y=x² and y=−2x².
4. Predict the turning point of y=(x+5)²−3.
5. Design a digital investigation of one quadratic parameter.
6. State a conjecture and a test case that could disprove it.

A student claims increasing any parameter always moves a graph upward. Use two function families to refute the claim and explain why parameter meaning depends on algebraic position.

Reasoning standard: Make a claim, show the relevant mathematical evidence, explain why it supports the conclusion and state any condition or limitation.

Why change one parameter at a time?

So the observed change can be attributed to that parameter.

What does m do in y=mx+c?

It controls gradient.

What follows a digital observation?

A generalisation supported by reasoning and tested cases.

1. [5 marks] Compare a family of linear graphs and infer roles of m and c.
2. [6 marks] Investigate y=a(x−h)² and generalise how a and h affect graph features.
3. [7 marks] Diagnose an apparent contradiction between a conjecture and graphing-tool output, then report a refined rule.

Review hint: A full-mark response shows the method, keeps units and restrictions visible, interprets the result in context and checks whether the answer is reasonable.

• I can vary one parameter systematically.
• I can predict graph changes before using software.
• I can connect changes to algebraic structure.
• I can test negative, zero and fractional cases.
• I can distinguish evidence from conjecture.

Exit ticket: Solve one unfamiliar example and explain the key decision in words, not just symbols.

Australian Curriculum v9.0 — AC9M9A06: experiment with the effects of the variation of parameters on graphs of related functions, using digital tools, making connections between graphical and algebraic representations, and generalising emerging patterns

Victoria: VC2M9A07 — Level 9 Algebra. The mapping names direct Level 9 content where available and explicitly identifies supporting content where the Victorian structure separates an idea differently.

NSW: Stage 5 Path — Functions and graphs, supported by Working mathematically. NSW organises Years 7–10 Mathematics through Stage 5 Core content groups and Paths rather than a one-code-per-Year-9 structure, so this lesson does not force a false one-to-one outcome.

Use these links to move through the Year 9 learning sequence rather than studying the code in isolation.

• AC9M9A03 — Gradient, Midpoint and Distance on the Cartesian Plane
• AC9M9A04 — Quadratic Functions, Graphs and Equations

• Australian Curriculum v9.0 — Mathematics
• Victorian Curriculum F–10 Version 2.0 — Mathematics
• NSW Mathematics K–10 Syllabus (2022)

Curriculum equivalents for Experiment with the effects of the variation of parameters on...

Mapped skill: experiment with the effects of the variation of parameters on graphs of related functions, using digital tools, making connections between graphical and algebraic representations, and generalising emerging patterns

These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.

Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.