Comparing linear/quadratic/exponential patterns | Year 10

01 / LEARN

Understand the idea.

Three different growth structures

At equal input intervals, test how the outputs change. A linear rule has constant first differences; a quadratic rule with a nonzero squared term has constant nonzero second differences; an exponential rule has a constant consecutive-output ratio when defined.

Compare $1, 3, 5, 7$ , whose differences are $2$ , with $1, 2, 4, 8$ , whose ratios are $2$ . Both increase, but one adds the same amount while the other multiplies by the same factor.

n	2n+1	n²	2ⁿ
0	1	0	1
1	3	1	2
2	5	4	4
3	7	9	8

Evidence is not a guarantee

Check the input spacing before using differences or ratios. A one-hour change and a three-hour change do not describe equal steps, so their raw ratios are not directly comparable in this test.

A short table supports a candidate model; it cannot prove the same pattern continues forever. Use the stated candidate families, the physical mechanism and further observations to judge whether a proposed continuation is reasonable.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Choose a family

Values at equal input steps are $5, 15, 45, 135$ . Which of linear, quadratic or exponential fits?

Worked solution

1
Test the consecutive ratios.
$\frac{15}{5} = 3, \frac{45}{15} = 3, \frac{135}{45} = 3$
The input intervals are equal and every output is nonzero.
2
Identify the fitting family.
$y = 5 \times 3^{n} (n = 0, 1, 2, 3)$
A constant multiplier of $3$ gives an exponential candidate; $n = 0$ labels the first supplied value.
3
Check the alternative difference patterns.
$\begin{aligned} first differences: & 10, 30, 90 \\ second differences: & 20, 60 \end{aligned}$
Neither the first nor second differences are constant, so the linear and quadratic families do not fit this table.

Answer

Exponential

EXAMPLE 02

Second differences

Values are $2, 5, 10, 17$ at $n = 0, 1, 2, 3$ . Compare the differences.

Worked solution

1
Calculate the first differences.
$5 - 2 = 3, 10 - 5 = 5, 17 - 10 = 7$
First differences compare adjacent outputs at equal input intervals.
2
Calculate the second differences.
$5 - 3 = 2, 7 - 5 = 2$
A constant nonzero second difference suggests a quadratic candidate.
3
Verify a matching quadratic rule.
$y = n^{2} + 2 n + 2$
Substituting $n = 0, 1, 2, 3$ gives $2, 5, 10, 17$ ; this fits the observations without proving a unique continuation.

Answer

Quadratic candidate

03 / YOUR TURN

Try it yourself.

Check your understanding

Classify $4, 9, 14, 19$ at equal steps.

Show answer & reasoning

Linear

The first differences are $9 - 4 = 14 - 9 = 19 - 14 = 5$ , so the values fit a linear pattern at the stated equal steps.

Keep practising

6 worksheet questions · 13 marks

Open worksheet Worked solutions

Read the questions online

Classify $3, 6, 9, 12$ at equal steps among linear, quadratic and exponential.
2 marks
Classify $2, 4, 8, 16$ at equal steps.
2 marks
Classify $1, 4, 9, 16$ at equal steps.
2 marks
Write an exponential rule for outputs $7, 14, 28, 56$ at $n = 0, 1, 2, 3$ .
2 marks
Find the next two values if $10, 20, 40$ continues with the same multiplicative rule.
2 marks
A model doubles daily. Why is adding $20$ each day not a valid general rule if it starts at $20$ ?
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Classify $3, 6, 9, 12$ at equal steps among linear, quadratic and exponential.
2 marks
All first differences are $3$ .
Linear
Classify $2, 4, 8, 16$ at equal steps.
2 marks
All consecutive ratios are $2$ .
Exponential
Classify $1, 4, 9, 16$ at equal steps.
2 marks
First differences are $3, 5, 7$ ; second differences are $2$ .
Quadratic
Write an exponential rule for outputs $7, 14, 28, 56$ at $n = 0, 1, 2, 3$ .
2 marks
The initial value is $7$ and the constant ratio is $2$ .
$y = 7 (2^{n})$
Find the next two values if $10, 20, 40$ continues with the same multiplicative rule.
2 marks
Multiply by $2$ at each step.
80 and 160
A model doubles daily. Why is adding $20$ each day not a valid general rule if it starts at $20$ ?
3 marks
Doubling gives $20, 40, 80, 160$ . Adding $20$ gives $20, 40, 60, 80$ , so agreement ends after the first step.
The daily increase itself grows.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum Mathematics

Integer and fractional index laws; connecting fractional powers with roots; simplifying surds; operations with surds; rationalising denominators; exact and approximate values; exponential tables and graphs; growth/decay factors; comparing linear/quadratic/exponential patterns; exponential equations; logarithms as inverses; logarithm laws; interpreting logarithmic scales; graph transformations and parameter experiments; choosing and evaluating models.

Official curriculum source ↗

In this lesson

Understand the idea.

Three different growth structures

Evidence is not a guarantee

See the method in action.

Choose a family

Test the consecutive ratios.

Identify the fitting family.

Check the alternative difference patterns.

Second differences

Calculate the first differences.

Calculate the second differences.

Verify a matching quadratic rule.

Try it yourself.

Check your understanding

Keep practising