Year 9 Mathematics · AC9M9A06

Parameter Variation and Graph Transformations

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

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Learning goalsSay it simply

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
Key conceptTeach from the board

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.

Worked examplesWe do

Worked examples

AC9M9A06 - Parameter Variation and Graph Transformations
Example 1

Vary intercept y=2x+1, y=2x+4 and y=2x−3 are parallel.

Example 2

Vary gradient y=x+2, y=3x+2 and y=−2x+2 share intercept but differ in steepness/direction.

Example 3

Vary quadratic scale y=x², 2x² and 0.5x² share turning point but differ in width.

Example 4

Shift a parabola y=(x−3)²+1 has turning point (3,1).

Curriculum examplesCopied content

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.

Lesson componentAustralian CurriculumVictoriaNSW
Concept teaching + worked examplesAC9M9A06VC2M9A07 — Level 9 AlgebraStage 5 Path — Functions and graphs, supported by Working mathematically
Guided + independent practiceApplies the descriptor through progressively less-scaffolded problemsBuilds the corresponding Level 9 mathematical knowledge and fluencySupports Stage 5 Core/Path application and Working mathematically
Reasoning + assessment + masteryChecks transfer, justification, interpretation and model limitsChecks Level 9 reasoning at the mapped content depthChecks relevant Stage 5 reasoning without claiming a false Year 9 equivalent
Questions and answersWith answers
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.
Practice and reviewReady for practice
  • 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. [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.

Curriculum alignmentStart here

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
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