Australian Curriculum v9 / ACiQ Year 9 Mathematics - Unit 2 - Parameters and related functions

Parameters and related functions

Experiment with parameter changes and connect graphical and algebraic representations of related functions.

Updated 2026-07-26 - 11 min read

Parameters and related functions 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.

Vary linear parameters

Changing m rotates the line about its intercept; changing c translates it vertically without changing gradient.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by vary linear parameters 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: Increasing c raises the y-intercept; steepness is controlled by m.

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

Compare y=2x+1 and y=2x+5.

Step 1 - identify the governing idea: In y=mx+c, m controls gradient and c controls the y-intercept.

Step 2 - apply it to this evidence: Both have m=2 and intercepts differ by 4.

Result: Same gradient; the second line is 4 units higher

The relationship is visible in the working: Both have m=2 and intercepts differ by 4. 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:

  • The second is 4 times steeper — It does not agree with the required relationship: Both have m=2 and intercepts differ by 4.
  • They are the same line — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The first has higher intercept — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

What changes from y=x−3 to y=−2x−3?

Step 1 - identify the governing idea: In y=mx+c, m controls gradient and c controls the y-intercept.

Step 2 - apply it to this evidence: Only m changes.

Result: Gradient changes from 1 to −2; intercept stays −3

The relationship is visible in the working: Only m changes. 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:

  • Only intercept changes — It does not agree with the required relationship: Only m changes.
  • Both gradients are −3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The lines are parallel — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Which line is parallel to y=4x+2?

Step 1 - identify the governing idea: In y=mx+c, m controls gradient and c controls the y-intercept.

Step 2 - apply it to this evidence: Parallel non-identical lines have the same gradient.

Result: y=4x−7

The relationship is visible in the working: Parallel non-identical lines have the same gradient. 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=−4x+2 — It does not agree with the required relationship: Parallel non-identical lines have the same gradient.
  • y=2x+4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • y=−7x+4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Vary quadratic parameters

Larger |a| makes the parabola narrower in the usual coordinate scale. Changing k moves the turning point vertically.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by vary quadratic parameters 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: a changes opening and stretch; k changes vertical position.

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

Compare y=x² and y=3x².

Step 1 - identify the governing idea: For y=ax²+k, the sign of a controls opening, |a| controls vertical stretch, and k shifts the graph vertically.

Step 2 - apply it to this evidence: The coefficient magnitude is larger.

Result: Both open up; y=3x² is narrower

The relationship is visible in the working: The coefficient magnitude is larger. 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:

  • The second shifts up 3 — It does not agree with the required relationship: The coefficient magnitude is larger.
  • The second opens down — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • They are parallel lines — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Compare y=x² and y=x²−5.

Step 1 - identify the governing idea: For y=ax²+k, the sign of a controls opening, |a| controls vertical stretch, and k shifts the graph vertically.

Step 2 - apply it to this evidence: k=−5 changes every y-value by −5.

Result: The second graph shifts down 5 units

The relationship is visible in the working: k=−5 changes every y-value by −5. 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:

  • It shifts right 5 — It does not agree with the required relationship: k=−5 changes every y-value by −5.
  • It becomes narrower — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It opens down — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

What does a=−1 do in y=−x²+4?

Step 1 - identify the governing idea: For y=ax²+k, the sign of a controls opening, |a| controls vertical stretch, and k shifts the graph vertically.

Step 2 - apply it to this evidence: A negative leading coefficient reflects it across the x-axis.

Result: The parabola opens downward

The relationship is visible in the working: A negative leading coefficient reflects it across the x-axis. 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:

  • It shifts left 1 — It does not agree with the required relationship: A negative leading coefficient reflects it across the x-axis.
  • It has no turning point — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • It opens upward — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Generalise parameter patterns

Testing several values can reveal a pattern, but edge cases such as zero and negative values must be included before generalising.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by generalise parameter patterns 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: Experiments suggest a conjecture; algebra and deliberate edge cases strengthen the justification.

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

A slider tests m=1,2,3 in y=mx. What extra case best tests the direction claim?

Step 1 - identify the governing idea: A digital pattern becomes mathematics when it is stated generally and connected to the algebraic role of the parameter.

Step 2 - apply it to this evidence: A negative case tests whether direction reverses.

Result: A negative value such as m=−2

The relationship is visible in the working: A negative case tests whether direction reverses. 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:

  • m=4 only — It does not agree with the required relationship: A negative case tests whether direction reverses.
  • c=10 in a different equation — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Remove axes — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

What happens to y=mx+c when c increases by d?

Step 1 - identify the governing idea: A digital pattern becomes mathematics when it is stated generally and connected to the algebraic role of the parameter.

Step 2 - apply it to this evidence: The equation adds d to every output.

Result: Every point's y-coordinate increases by d

The relationship is visible in the working: The equation adds d to every output. 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:

  • Every x-coordinate increases by d — It does not agree with the required relationship: The equation adds d to every output.
  • Gradient increases by d — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Only the intercept point moves — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Best generalisation for y=ax² when |a| increases?

Step 1 - identify the governing idea: A digital pattern becomes mathematics when it is stated generally and connected to the algebraic role of the parameter.

Step 2 - apply it to this evidence: All non-zero outputs are multiplied by a larger magnitude.

Result: For non-zero a, the graph is vertically stretched and appears narrower

The relationship is visible in the working: All non-zero outputs are multiplied by a larger magnitude. 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:

  • The vertex moves right — It does not agree with the required relationship: All non-zero outputs are multiplied by a larger magnitude.
  • The graph always opens upward — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • The graph becomes a line — 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. What changes from y=x−3 to y=−2x−3?
  2. Compare y=x² and y=x²−5.
  3. What happens to y=mx+c when c increases by d?

Answers

  1. Gradient changes from 1 to −2; intercept stays −3 — Only m changes.
  2. The second graph shifts down 5 units — k=−5 changes every y-value by −5.
  3. Every point's y-coordinate increases by d — The equation adds d to every output.

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