Comparing linear/quadratic/exponential patterns | Year 10

01 / LEARN

The idea, step by step

Three different growth structures

A linear pattern has constant first differences at equal input steps. A quadratic pattern has constant second differences. An exponential pattern has a constant ratio of consecutive outputs when those outputs are nonzero.

For inputs $0, 1, 2, 3$ , compare $2 n + 1$ : $1, 3, 5, 7$ ; $n^{2}$ : $0, 1, 4, 9$ ; and $2^{n}$ : $1, 2, 4, 8$ . Their rules describe different kinds of change.

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

A short table can be fitted by many rules. A constant ratio suggests an exponential model within the candidate families, but context and additional observations are needed.

Check that the input intervals are equal. A ratio comparison between one-hour and three-hour intervals is not the same test.

A common trap

A growing set of numbers must be exponential.

Instead: Linear, quadratic and many other rules can increase. Test the differences and ratios, then consider the context.

02 / WORKED EXAMPLES

See the thinking

Example 1

Choose a family

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

Consecutive ratios are all $3$ .
An exponential model with initial value $5$ is $y = 5 (3^{n})$ for $n = 0, 1, 2, 3$ .

Exponential

Example 2

Second differences

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

First differences are $3, 5, 7$ .
Second differences are $2, 2$ , suggesting a quadratic model.
The rule $y = n^{2} + 2 n + 2$ fits these values.

Quadratic candidate

Check your understanding

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

Show answer

Linear

The first difference is always $5$ .

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF 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 ↗

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