Exponential graphs and constant percentage change | Year 11

01 / LEARN

The idea, step by step

A fixed multiplier

In $P = A b^{n}$ , A is the value at $n = 0$ and b is the multiplier for each equal input step. Consecutive positive outputs have ratio b. A value $b > 1$ gives growth; a value $0 < b < 1$ gives decay. If $b = 1$ , the quantity stays constant.

The multiplier retains the original whole as well as any increase. Thus 1.05 means 5% growth. A factor 0.80 retains 80%, meaning a 20% decrease, not an 80% decrease.

P = A b^{n}

The shape and the interval

An exponential growth graph with positive starting amount bends upward. The absolute increases become larger even though the percentage is constant. A simple exponential decay model remains positive while its values decrease.

The exponent counts intervals. An annual factor must be used with years, and a monthly factor with months. A continuous curve between discrete observations is an additional modelling assumption; keep that distinction clear.

A common trap

Reading a factor of 0.8 as an 80% decrease.

Instead: The factor is the proportion remaining. Subtract it from 1 to find the decrease.

02 / WORKED EXAMPLES

See the thinking

Example 1

Consecutive annual values

An amount follows $P = 800 {(1.05)}^{n}$ . Find years 0 to 3.

Year	0	1	2	3
Amount	800	840	882	926.10

At zero the amount is 800.
Multiply successively by 1.05: 840, 882, 926.10.
Increases are 40, 42 and 44.10, although all represent 5% of the previous value.

800, 840, 882 and 926.10.

Example 2

A decay factor

Interpret 0.8 in $V = 1200 {(0.8)}^{t}$ and find $V (2)$ .

80% remains, so 20% is removed per interval.
At 2, $V = 1200 {(0.8)}^{2} = 768$ .
The second decrease is based on 960, not the original 1200.

20% decrease per interval;

V (2) = 768

Check your understanding

What change is represented by factor 1.12?

Show answer

12% growth per interval.

$1.12 - 1 = 0.12 = 12 %$ .

03 / WORKSHEET

Your turn

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

Worksheet PDF Worked solutions

Read the questions online

Write the multiplier for 7% growth.
2 marks
Write the multiplier for 15% decay.
2 marks
Values are 50, 60, 72, 86.4 at equal steps. Find factor and percentage change.
2 marks
Evaluate $300 {(1.1)}^{2}$ .
2 marks
Evaluate $500 {(0.9)}^{3}$ .
2 marks
Explain why 20, 25, 30, 35 is not exponential with one constant factor.
3 marks

Read the worked solutions

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

Write the multiplier for 7% growth.
2 marks
$1 + 0.07 = 1.07$ .
1.07.
Write the multiplier for 15% decay.
2 marks
$1 - 0.15 = 0.85$ .
0.85.
Values are 50, 60, 72, 86.4 at equal steps. Find factor and percentage change.
2 marks
Every consecutive ratio is 1.2.
The excess above 1 is 0.2.
Factor 1.2; 20% growth.
Evaluate $300 {(1.1)}^{2}$ .
2 marks
${1.1}^{2} = 1.21$ ; multiply by 300.
363.
Evaluate $500 {(0.9)}^{3}$ .
2 marks
${0.9}^{3} = 0.729$ ; multiply by 500.
364.5.
Explain why 20, 25, 30, 35 is not exponential with one constant factor.
3 marks
Ratios are 1.25, 1.2 and about 1.1667.
Its constant difference 5 instead indicates linear change.
Its consecutive ratios differ.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: algebraic and graphical modelling of linear/exponential change, with interpretation and evaluation.

Official curriculum source ↗

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