Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 2 - Quadratic graphs and equations

Quadratic graphs and equations

Identify and graph quadratic functions and solve monic quadratics graphically, numerically and algebraically.

Updated 2026-07-26 - 10 min read

Quadratic graphs and equations is part of the Year 9 curriculum because students must do more than carry out a familiar calculation. They need to choose a relationship, represent it accurately, explain the result and decide whether it makes sense. The sections below develop those decisions through explicit rules, worked examples, misconception repair and transfer.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Identify and graph quadratics

A value table should include points on both sides of the axis of symmetry. Connecting too few points with straight segments hides the curved shape.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by identify and graph quadratics and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The highest non-zero power of x must be 2.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Which is quadratic: y=3x+2, y=x²−4, y=2/x or y=x³?

Step 1 - identify the governing idea: A quadratic function has degree 2 and graphs as a parabola.

Step 2 - apply it to this evidence: Its highest power of x is 2.

Result: y=x²−4

The relationship is visible in the working: Its highest power of x is 2. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • y=3x+2 — It does not agree with the required relationship: Its highest power of x is 2.
  • y=2/x — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • y=x³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

For y=x²−4, what are the values at x=−2,0,2?

Step 1 - identify the governing idea: A quadratic function has degree 2 and graphs as a parabola.

Step 2 - apply it to this evidence: Substitution gives symmetry about x=0.

Result: 0, −4, 0

The relationship is visible in the working: Substitution gives symmetry about x=0. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • −8, −4, 8 — It does not agree with the required relationship: Substitution gives symmetry about x=0.
  • 4, −4, 4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0, 4, 0 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

What shape should y=−x²+3 have?

Step 1 - identify the governing idea: A quadratic function has degree 2 and graphs as a parabola.

Step 2 - apply it to this evidence: The negative leading coefficient reflects the parabola downward.

Result: A downward-opening parabola

The relationship is visible in the working: The negative leading coefficient reflects the parabola downward. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • An upward-opening parabola — It does not agree with the required relationship: The negative leading coefficient reflects the parabola downward.
  • A straight line — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A circle — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Solve quadratics from roots

If a product of factors equals zero, at least one factor must be zero. This zero-product property links algebraic factors to graphical intercepts.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by solve quadratics from roots and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Solutions are x-intercepts of y=f(x); the turning point is a different feature unless it lies on the x-axis.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Solve (x−2)(x+5)=0.

Step 1 - identify the governing idea: Solutions of f(x)=0 are the x-values where the graph crosses or touches the x-axis.

Step 2 - apply it to this evidence: Each factor can equal zero.

Result: x=2 or x=−5

The relationship is visible in the working: Each factor can equal zero. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x=−2 or x=5 — It does not agree with the required relationship: Each factor can equal zero.
  • x=3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x=−10 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A parabola crosses the x-axis at −1 and 4. Solve f(x)=0.

Step 1 - identify the governing idea: Solutions of f(x)=0 are the x-values where the graph crosses or touches the x-axis.

Step 2 - apply it to this evidence: The x-intercepts are the roots.

Result: x=−1 or x=4

The relationship is visible in the working: The x-intercepts are the roots. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x=3 — It does not agree with the required relationship: The x-intercepts are the roots.
  • x=1.5 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • y=−1 or y=4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Solve x²−9=0.

Step 1 - identify the governing idea: Solutions of f(x)=0 are the x-values where the graph crosses or touches the x-axis.

Step 2 - apply it to this evidence: x²−9=(x−3)(x+3).

Result: x=−3 or x=3

The relationship is visible in the working: x²−9=(x−3)(x+3). Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x=9 — It does not agree with the required relationship: x²−9=(x−3)(x+3).
  • x=3 only — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x=−9 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Factorise and solve monic quadratics

Checking by substitution detects sign errors and confirms both roots satisfy the original equation.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by factorise and solve monic quadratics and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Set each factor equal to zero because their product is zero.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Solve x²+5x+6=0.

Step 1 - identify the governing idea: For x²+bx+c=0, find integers m,n with m+n=b and mn=c, then solve (x+m)(x+n)=0.

Step 2 - apply it to this evidence: (x+2)(x+3)=0.

Result: x=−2 or x=−3

The relationship is visible in the working: (x+2)(x+3)=0. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x=2 or x=3 — It does not agree with the required relationship: (x+2)(x+3)=0.
  • x=−1 or x=−6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x=6 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Solve x²−7x+12=0.

Step 1 - identify the governing idea: For x²+bx+c=0, find integers m,n with m+n=b and mn=c, then solve (x+m)(x+n)=0.

Step 2 - apply it to this evidence: (x−3)(x−4)=0.

Result: x=3 or x=4

The relationship is visible in the working: (x−3)(x−4)=0. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x=−3 or x=−4 — It does not agree with the required relationship: (x−3)(x−4)=0.
  • x=2 or x=6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x=12 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Solve x²+2x−15=0.

Step 1 - identify the governing idea: For x²+bx+c=0, find integers m,n with m+n=b and mn=c, then solve (x+m)(x+n)=0.

Step 2 - apply it to this evidence: (x−3)(x+5)=0.

Result: x=3 or x=−5

The relationship is visible in the working: (x−3)(x+5)=0. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x=−3 or x=5 — It does not agree with the required relationship: (x−3)(x+5)=0.
  • x=1 or x=−15 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x=−2 or x=15 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. For y=x²−4, what are the values at x=−2,0,2?
  2. A parabola crosses the x-axis at −1 and 4. Solve f(x)=0.
  3. Solve x²−7x+12=0.

Answers

  1. 0, −4, 0 — Substitution gives symmetry about x=0.
  2. x=−1 or x=4 — The x-intercepts are the roots.
  3. x=3 or x=4 — (x−3)(x−4)=0.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources