Quadratic Functions, Graphs and Equations — AC9M9A04

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

Be comfortable expanding and factorising monic quadratics, plotting coordinates, reading graphs and substituting values.

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.

• 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. For y=x²−4x+3, factor and predict x-intercepts before graphing.
2. Solve x²−9=0 and explain both roots.
3. From a parabola, identify axis of symmetry, turning point and roots.
4. Use a digital graph to estimate roots of x²−2=0.

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. Factor and solve x²+7x+12=0.
2. Solve x²−2x−15=0.
3. Explain why x²+4=0 has no real roots.
4. Create a value table for y=x²−4 and identify second differences.
5. For y=(x−1)(x+5), state roots without expanding.
6. Compare an exact root and graphically estimated root for the same equation.

A graphing window shows only one x-intercept of a quadratic that algebraically has two real roots. Explain how the window can mislead and describe a systematic check.

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

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.

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.

• I can connect a quadratic rule, table and graph.
• I can interpret roots as x-intercepts.
• I can solve monic quadratics with integer roots by factorisation.
• I remember both solutions to x²=a when a>0.
• I can use digital tools to estimate and verify, not guess.

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

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.

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

• AC9M9A02 — Algebraic Expressions, Binomial Expansion and Factorisation
• AC9M9A06 — Parameter Variation and Graph Transformations

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

Curriculum equivalents for And graph quadratic functions, solve quadratic equations graphically and numerically...

Mapped skill: 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

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.