Exponential graphs and constant percentage change | Year 11

01 / LEARN

Understand the idea.

A fixed multiplier

In $P = A b^{n}$ , the starting amount is $A$ because $b^{0} = 1$ . Each equal input step multiplies the current value by $b$ . For positive $A$ , growth has $b > 1$ and decay has $0 < b < 1$ .

The factor includes the original whole. A factor of $1.05$ keeps the whole and adds five percent; a factor of $0.80$ keeps eighty percent and removes twenty percent. A factor of one gives no change.

P = A b^{n}

The shape and the interval

Exponential change keeps a percentage, rather than an absolute increment, constant. During positive growth, later increases become larger because the same percentage is taken of a larger amount. During decay, later decreases become smaller.

The exponent counts the intervals attached to the factor. Annual factors require years and monthly factors require months. Connecting integer-time observations with a smooth curve assumes the exponential rule also applies between those observations.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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

Worked solution

1
Evaluate the starting amount.
$P (0) = 800 {(1.05)}^{0} = 800$
A zero exponent gives a multiplier of one.
2
Apply the annual factor twice.
$\begin{aligned} P (1) & = 800 (1.05) = 840 \\ P (2) & = 840 (1.05) = 882 \end{aligned}$
Each year uses the previous year’s amount as its base.
3
Apply the factor for year three.
$P (3) = 882 (1.05) = 926.10$
The third growth step is five percent of eight hundred and eighty-two.
4
Compare the absolute increases.
$\begin{aligned} 840 - 800 & = 40 \\ 882 - 840 & = 42 \\ 926.10 - 882 & = 44.10 \end{aligned}$
The increments differ although each is five percent of its preceding amount.

Answer

800, 840, 882 and 926.10.

EXAMPLE 02

A decay factor

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

Worked solution

1
Interpret the remaining fraction.
$0.8 = 80 %, 100 % - 80 % = 20 %$
The factor states how much remains, so the removed percentage is its complement.
2
Calculate the first interval.
$V (1) = 1200 (0.8) = 960$
The first reduction removes twenty percent of twelve hundred.
3
Calculate the second interval.
$V (2) = 960 (0.8) = 768$
The second reduction acts on nine hundred and sixty, not on the original value.

Answer

20% decrease per interval; $V (2) = 768$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

What change is represented by factor 1.12?

Show answer & reasoning

12% growth per interval.

The amount retained is one whole plus $0.12$ : $1.12 - 1 = 0.12 = 12 %$ . This is twelve-percent growth per interval.

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

A fixed multiplier

The shape and the interval

See the method in action.

Consecutive annual values

Evaluate the starting amount.

Apply the annual factor twice.

Apply the factor for year three.

Compare the absolute increases.

A decay factor

Interpret the remaining fraction.

Calculate the first interval.

Calculate the second interval.

Try it yourself.

Check your understanding

Keep practising