Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 1 - Exponential relations and equations

Exponential relations and equations

Recognise, represent and solve exponential relations in which equal input steps multiply outputs by a constant factor.

Updated 2026-07-26 - 11 min read

Exponential relations and equations is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

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.

Recognise exponential patterns

An exponential pattern has a constant multiplicative ratio for equal input intervals. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by recognise exponential 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: Growth speed alone is insufficient; consecutive outputs must share a constant factor.

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 sequence is exponential?

Step 1 - identify the governing idea: An exponential pattern has a constant multiplicative ratio for equal input intervals.

Step 2 - apply it to this evidence: Each term is multiplied by 2.

Result: 3, 6, 12, 24

The relationship is visible in the working: Each term is multiplied by 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:

  • 3, 6, 9, 12 — It does not agree with the required relationship: Each term is multiplied by 2.
  • 2, 5, 10, 17 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4, 4, 4, 4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

For y = 5(3ˣ), what factor changes y when x increases by 1?

Step 1 - identify the governing idea: An exponential pattern has a constant multiplicative ratio for equal input intervals.

Step 2 - apply it to this evidence: The exponential base 3 is the repeated multiplier.

Result: 3

The relationship is visible in the working: The exponential base 3 is the repeated multiplier. 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:

  • 5 — It does not agree with the required relationship: The exponential base 3 is the repeated multiplier.
  • 8 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 15 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A population doubles every hour from 40. Which rule fits after h hours?

Step 1 - identify the governing idea: An exponential pattern has a constant multiplicative ratio for equal input intervals.

Step 2 - apply it to this evidence: The initial value is 40 and the repeated factor is 2.

Result: P = 40(2ʰ)

The relationship is visible in the working: The initial value is 40 and the repeated factor 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:

  • P = 2(40ʰ) — It does not agree with the required relationship: The initial value is 40 and the repeated factor is 2.
  • P = 40 + 2h — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • P = 80h — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Evaluate and solve simple exponential relations

Substitute inputs using operation order and solve simple equations by expressing both sides with a common base or testing integer exponents. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by evaluate and solve simple exponential relations 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 exponent counts repeated multiplication; 16 = 2⁴, so x = 4.

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

Evaluate y = 3(2ˣ) when x = 4.

Step 1 - identify the governing idea: Substitute inputs using operation order and solve simple equations by expressing both sides with a common base or testing integer exponents.

Step 2 - apply it to this evidence: 2⁴ = 16 and 3 × 16 = 48.

Result: 48

The relationship is visible in the working: 2⁴ = 16 and 3 × 16 = 48. 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:

  • 24 — It does not agree with the required relationship: 2⁴ = 16 and 3 × 16 = 48.
  • 81 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 14 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Solve 5ˣ = 125.

Step 1 - identify the governing idea: Substitute inputs using operation order and solve simple equations by expressing both sides with a common base or testing integer exponents.

Step 2 - apply it to this evidence: 125 = 5³.

Result: 3

The relationship is visible in the working: 125 = 5³. 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:

  • 25 — It does not agree with the required relationship: 125 = 5³.
  • 5 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Solve 2ˣ⁺¹ = 32.

Step 1 - identify the governing idea: Substitute inputs using operation order and solve simple equations by expressing both sides with a common base or testing integer exponents.

Step 2 - apply it to this evidence: 32 = 2⁵, so x + 1 = 5 and x = 4.

Result: 4

The relationship is visible in the working: 32 = 2⁵, so x + 1 = 5 and x = 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:

  • 5 — It does not agree with the required relationship: 32 = 2⁵, so x + 1 = 5 and x = 4.
  • 16 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 31 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Connect exponential tables, rules and graphs

Read the initial value, repeated factor and domain from a rule or table and use them to explain graph shape and context. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by connect exponential tables, rules and graphs 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: At x = 0, b⁰ = 1, so the intercept of a bˣ is a.

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

For y = 12(0.5ˣ), what is y when x = 0?

Step 1 - identify the governing idea: Read the initial value, repeated factor and domain from a rule or table and use them to explain graph shape and context.

Step 2 - apply it to this evidence: 0.5⁰ = 1, so y = 12.

Result: 12

The relationship is visible in the working: 0.5⁰ = 1, so y = 12. 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:

  • 0.5 — It does not agree with the required relationship: 0.5⁰ = 1, so y = 12.
  • 6 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

Outputs are 80, 40, 20, 10 for x = 0,1,2,3. Which rule fits?

Step 1 - identify the governing idea: Read the initial value, repeated factor and domain from a rule or table and use them to explain graph shape and context.

Step 2 - apply it to this evidence: The initial value is 80 and each step multiplies by 0.5.

Result: y = 80(0.5ˣ)

The relationship is visible in the working: The initial value is 80 and each step multiplies by 0.5. 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:

  • y = 0.5(80ˣ) — It does not agree with the required relationship: The initial value is 80 and each step multiplies by 0.5.
  • y = 80 − 40x — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • y = 80(2ˣ) — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

What feature identifies exponential decay in y = a bˣ for a > 0?

Step 1 - identify the governing idea: Read the initial value, repeated factor and domain from a rule or table and use them to explain graph shape and context.

Step 2 - apply it to this evidence: Repeated multiplication by a positive fraction reduces the output.

Result: 0 < b < 1

The relationship is visible in the working: Repeated multiplication by a positive fraction reduces the 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:

  • b > 1 — It does not agree with the required relationship: Repeated multiplication by a positive fraction reduces the output.
  • b = 0 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • a < 0 only — 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 = 5(3ˣ), what factor changes y when x increases by 1?
  2. Solve 5ˣ = 125.
  3. Outputs are 80, 40, 20, 10 for x = 0,1,2,3. Which rule fits?

Answers

  1. 3 — The exponential base 3 is the repeated multiplier.
  2. 3 — 125 = 5³.
  3. y = 80(0.5ˣ) — The initial value is 80 and each step multiplies by 0.5.

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