KS3 Maths · Year 8 · Chapter 5: Applications of graphs · Lesson 3

Exponential growth graphs

Year 8About 70 minutesCalculator allowed45 marks

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Quick questions to start.

A1
Last lesson
A bath fills at $8$ litres a minute. How long does $52$ litres take?
[1 mark]
Answer
A2
Needed today
Work out $3 \times 2^{4}$ .
[1 mark]
Answer
A3
Needed today
Write down the multiplier for an increase of $6%$ .
[1 mark]
Answer
A4
From chapter 1
Increase £ $400$ by $10%$ , then by $10%$ again.
[1 mark]
Answer

Example, then your turn

Year 8

Your teacher works through each example with you. Then try the one beside it.

Example 1

A colony of $300$ bees doubles in size every $3$ months. (a) Complete a table of the number of bees after $0$ , $3$ , $6$ , $9$ and $12$ months. (b) The points are plotted and joined with a smooth curve. Estimate when there are $3000$ bees.

Hint Double each number to get the next one. To read a time, go across from

3000

to the curve, then down.

B1
A lake has $400$ fish. The number doubles every $2$ years. (a) Complete a table for $0$ , $2$ , $4$ , $6$ and $8$ years. (b) Plot the points on the grid and join them with a smooth curve. (c) Estimate when there are $5000$ fish.
[4 marks]

Example 2

The number of bacteria, $n$ , in a dish after $t$ hours is $n = 48 \times 1. 5^{t}$ . (a) Complete a table of $n$ for $t = 0$ , $1$ , $2$ , $3$ and $4$ . (b) The graph of $n = 48 \times 1. 5^{t}$ is drawn. Estimate the number of bacteria after $2 \frac{1}{2}$ hours. (c) Estimate when there are $200$ bacteria.

Hint Put each value of

t

into the formula with your calculator:

48 \times 1. 5^{3} = 162

. Read the graph up from

2.5

, and across from

200

B2
The number of views of a video, $n$ thousand, after $t$ days is $n = 25 \times 1. 8^{t}$ . (a) Complete a table of $n$ for $t = 0$ , $1$ , $2$ , $3$ and $4$ . Give each value to the nearest whole number. (b) The graph is drawn. Use it to estimate when the video has $200$ thousand views.
[3 marks]

Example 3

£ $500$ is invested at $6%$ compound interest a year. The amount after $n$ years is £ $A$ , where $A = 500 \times 1.0 6^{n}$ . (a) Work out the amount after $5$ years. (b) After how many complete years has the amount first doubled?

Hint Doubled means £

1000

or more. Try values of

n

with your calculator, and look for the first year that goes past £

1000

B3
£ $800$ is invested at $9%$ compound interest a year. The amount after $n$ years is £ $A$ , where $A = 800 \times 1.0 9^{n}$ . (a) Work out the amount after $4$ years. (b) After how many complete years has the amount first doubled?
[4 marks]

Practice

Year 8

C1
Which of these tables show exponential growth? Give a reason. P: $8$ , $19$ , $30$ , $41$ , $52$ . Q: $11$ , $33$ , $99$ , $297$ , $891$ . R: $8$ , $40$ , $200$ , $1000$ , $5000$ . S: $10$ , $12$ , $16$ , $22$ , $30$ .
[2 marks]
C2
A table for $y = a \times b^{x}$ is: $x = 0$ , $1$ , $2$ , $3$ and $y = 11$ , $44$ , $176$ , $704$ . Find $a$ and $b$ . Work out $y$ when $x = 5$ .
[3 marks]
Answer $a = \dots$ $b = \dots$ $y = \dots$

C3
The graph shows the mass of a medicine left in a patient's body. The half-life is the time for the mass to fall to half of what it was at the start. Use the graph to estimate (i) the half-life, (ii) the mass left after $5$ days.
[2 marks]
C4
A population, $P$ , doubles every $4$ years. After $t$ years, $P = 300 \times 2^{\frac{t}{4}}$ . Work out $P$ when $t = 12$ , and when $t = 10$ . Give your second answer to the nearest whole number.
[3 marks]
Answer $t = 12$ : $\dots$ $t = 10$ : $\dots$

Exam-style questions

GCSE Foundation

D1
Mia says, "If something grows by $20%$ every year, it will have doubled after $5$ years, because $5 \times 20% = 100%$ ."
Is Mia correct? Show working to support your answer.
[3 marks]

D2
The number of cells, $n$ , in a sample after $t$ hours is $n = 160 \times 1. 5^{t}$ .
(a)
Complete a table of $n$ for $t = 0$ , $1$ , $2$ , $3$ , $4$ and $5$ .
[2 marks]
(b)
Draw the graph of $n = 160 \times 1. 5^{t}$ on the grid.
[2 marks]
(c)
Use your graph to estimate when there are $1000$ cells.
[1 mark]
Answer $\dots$ hours
Total for D2: 5 marks
D3
The population of Town P goes up by $200$ people each year. The population of Town Q goes up by $20%$ each year. The graph shows both populations.
(a)
Which graph, A or B, shows Town Q? Give a reason.
[1 mark]
(b)
Use the graph to estimate when the population of Town Q becomes bigger than the population of Town P.
[1 mark]
Answer $\dots$ years
(c)
The population of Town Q was $1000$ at the start. Work out its population after $8$ years. Give your answer to the nearest whole number.
[2 marks]
Answer
Total for D3: 4 marks
D4
A patch of weed on a pond covers $3$ m². The area it covers doubles every week. The pond has an area of $96$ m².
(a)
After how many weeks does the weed cover the whole pond?
[2 marks]
Answer $\dots$ weeks
(b)
Ravi says, "The pond is half covered after $2 \frac{1}{2}$ weeks." Explain why Ravi is wrong.
[2 marks]
Total for D4: 4 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The number of cells in a dish is multiplied by the same number every hour. There are $50$ cells at 9:00 am and $450$ cells at 11:00 am. Estimate how many cells there were at 10:30 am.
[4 marks]
Hint 1 · What to try
Two hours multiply the number by $450 \div 50 = 9$ . What is the multiplier for one hour?
Hint 2 · The first line
One hour multiplies by $3$ , so at 10:00 am there were $150$ cells. Half an hour multiplies by a number which, done twice, gives $3$ .
Hint 3 · The full method
Half an hour multiplies by $3 = 1.732 \dots$ , so at 10:30 am there were $150 \times 3 = 259.8$ , about $260$ cells.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can make a table for an amount that doubles, or is multiplied by the same number, each time.
I can draw an exponential graph and read values from it, forwards and backwards.
I can use $y = a \times b^{x}$ for growth, decay and compound interest.
I can tell exponential growth from growth by the same amount each time.

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