Year 9 Mathematics · AC9M9A04

Quadratic Functions, Graphs and Equations

Quadratic rules, tables, parabolas, factors and roots are different representations of the same relationship. Year 9 connects graphical, numerical and algebraic solutions

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Learning goalsSay it simply

Quadratic rules, tables, parabolas, factors and roots are different representations of the same relationship. Year 9 connects graphical, numerical and algebraic solutions.

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

  • identify quadratic patterns and graph quadratic functions
  • interpret turning points, symmetry and intercepts
  • solve quadratic equations graphically and numerically
  • solve monic quadratics with integer roots using factors and the null factor law
Key conceptTeach from the board

Quadratic graphs are parabolas

A quadratic such as y=x²−5x+6 has a curved graph with an axis of symmetry and turning point.

Roots and x-intercepts are the same solution idea

Solving x²−5x+6=0 asks where y=0. Graphically these are x-intercepts; algebraically (x−2)(x−3)=0 gives x=2 or 3.

Numerical and graphical solutions may be approximate

Digital tools can locate roots when exact factorisation is not immediate. State the accuracy of approximate roots.

Features connect across representations

Tables can show constant second differences; factorised form exposes roots; graphs expose symmetry and turning points.

Worked examplesWe do

Worked examples

AC9M9A04 - Quadratic Functions, Graphs and Equations
Example 1

Recognise a quadratic table For y=x², first differences change but second differences are constant.

Example 2

Factor and solve x²−7x+12=0 → (x−3)(x−4)=0 → x=3 or 4.

Example 3

No real roots If a parabola never crosses the x-axis, the corresponding equation has no real roots.

Example 4

Two square-root solutions x²=20 gives x=±√20=±2√5≈±4.472.

Curriculum examplesCopied content

Australian Curriculum v9.0 — AC9M9A04: identify and graph quadratic functions, solve quadratic equations graphically and numerically, and solve monic quadratic equations with integer roots algebraically, using graphing software and digital tools as appropriate

Victoria: VC2M9A05 — 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 Core — Linear and non-linear relationships; Paths — Further algebra and equations / Functions and graphs. 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 examplesAC9M9A04VC2M9A05 — Level 9 AlgebraStage 5 Core — Linear and non-linear relationships; Paths — Further algebra and equations / Functions and graphs
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
What is a root?
An x-value that makes the quadratic equal to zero.
How are roots shown on a graph?
As x-intercepts.
Why use different methods?
Factorisation can give exact roots; graphs and numerical tools show meaning and approximate non-integer roots.
Practice and reviewReady for practice
  • A root is the y-intercept: Roots correspond to x-intercepts, where y=0.
  • x²=25 has only x=5: Both 5 and −5 work.
  • Every quadratic factorises over integers: Some require numerical or graphical approximation at this level.
  • Graph and equation are separate topics: They represent the same relationship and can verify each other.
  1. [4 marks] Factorise and solve two monic quadratic equations with integer roots.
  2. [5 marks] From a supplied quadratic graph, determine roots, turning point, symmetry and where the function is negative.
  3. [7 marks] Verify two approximate digital roots using algebraic structure and report them to stated accuracy.

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

Quadratic rules, tables, parabolas, factors and roots are different representations of the same relationship. Year 9 connects graphical, numerical and algebraic solutions.

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

  • identify quadratic patterns and graph quadratic functions
  • interpret turning points, symmetry and intercepts
  • solve quadratic equations graphically and numerically
  • solve monic quadratics with integer roots using factors and the null factor law
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