GCSE Maths · Higher · Chapter 23: Graphs · Lesson 4

Cubic, reciprocal and exponential graphs

Grades 5–8About 50 minutesCalculator allowed41 marks

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NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Where does $y = (x - 6) (x + 4)$ cross the $x$ -axis?
[1 mark]
Answer
A2
From chapter 23
One trapezium strip has width 4 and heights 8 and 14. Find its area.
[1 mark]
Answer
A3
From chapter 8
Simplify $a^{3} \times a^{4}$ .
[1 mark]
Answer
A4
From chapter 15
Solve $4 x + 1 < 13$ .
[1 mark]
Answer

Grades 5–7

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

Example 1

The curve $y = a b^{x}$ passes through $(0, 6)$ and $(3, 48)$ . Find $a$ and $b$ .

Hint Put

x = 0

in first:

b^{0} = 1

. Then use the second point.

B1
The curve $y = a b^{x}$ passes through $(0, 4)$ and $(2, 100)$ . Find $a$ and $b$ , then work out $y$ when $x = 3$ .
[3 marks]
Answer $a = \dots\dots$ $b = \dots\dots$ $y = \dots\dots$

Example 2

For the curve $y = (x + 1) (x - 2) (x - 4)$ , find where it crosses the $x$ -axis and where it crosses the $y$ -axis.

Hint

y = 0

when any one bracket is 0. Put

x = 0

in for the

y

-axis.

B2
For the curve $y = (x + 3) (x - 1) (x - 5)$ , find where it crosses the $x$ -axis and where it crosses the $y$ -axis.
[3 marks]
Answer $x = \dots\dots$ , $\dots\dots$ , $\dots\dots$ $(0, \dots\dots)$

Grades 5–7

C1
The point $(x, 4.5)$ lies on the curve $y = \frac{54}{x}$ . Work out $x$ .
[2 marks]
Answer
C2
Work out the value of $y = 6^{x}$ when $x = - 2$ . Give your answer as a fraction.
[2 marks]
Answer

C3
The sketch shows a curve. Which of $y = x^{3} - 7 x$ , $y = \frac{7}{x}$ and $y = 7^{x}$ could it be?
[1 mark]
Answer
C4
For $y = x^{3} - 5 x + 2$ , work out $y$ when $x = - 3$ .
[2 marks]
Answer
C5
The curve $y = \frac{36}{x}$ and the line $y = 4 x$ meet at two points. Work out the coordinates of both points.
[3 marks]
Answer $(\dots, \dots)$ and $(\dots, \dots)$
C6
A car is worth £18000. Its value after $t$ years is $V = 18000 \times 0.8 8^{t}$ . Work out its value after 3 years, to the nearest pound.
[2 marks]
Answer£ $\dots\dots$
C7
The graph of $y = 2^{x}$ is drawn. Use the graph to estimate the solution of $2^{x} = 6$ .
[2 marks]
Answer $x = \dots\dots$
C8
The curve $y = \frac{k}{x}$ passes through the point $(- 2, 5)$ . Find $k$ . In which two quadrants does the curve lie?
[2 marks]
Answer $k = \dots\dots$ quadrants $\dots\dots$

Grades 6–8

D1
Owen says, "The curve $y = \frac{12}{x}$ crosses the $y$ -axis at $(0, 12)$ ."
Explain why Owen is wrong.
[2 marks]

D2
A lake is stocked with fish. The number of fish, $N$ , $t$ years after the first count is modelled by $N = a b^{t}$ . After 1 year there are 800 fish. After 3 years there are 1250 fish.
(a)
Show that $b = 1.25$
[2 marks]
(b)
Work out the number of fish at the first count.
[2 marks]
Answer
(c)
Use the model to work out the number of fish after 5 years. Give your answer to the nearest whole number.
[2 marks]
Answer
Total for D2: 6 marks
D3
The curve $y = 2 x^{3} - 50 x$ crosses the $x$ -axis at three points. Work out the $x$ -coordinate of each point. You must show your working.
[3 marks]
Answer $x = \dots\dots$ , $\dots\dots$ and $\dots\dots$

Grade 8 and beyond

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out values on cubic, reciprocal and exponential curves.
I can tell those three kinds of graph apart.
I can find $a$ and $b$ for $y = a b^{x}$ from two points.

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