Drawing Other Graphs: Cubic/Reciprocal worksheet

7 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

7 questions 22 marks Sheet 1

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3 marks The table shows some values of \(y = x^3 - 3x + 4\)

x -3 -2 -1 0 1 2 3

y -14 6 4 6

Three of the \(y\) values are missing.
(a)
Work out the value of \(y\) when \(x = -2\).

(b)
Work out the value of \(y\) when \(x = 1\).

(c)
Work out the value of \(y\) when \(x = 3\).
2 marks
The graph of \(y = \frac{ 8 }{ x }\) is drawn on the grid. Each square is \(1\).

The graph is in two separate pieces, one in each of two opposite corners.

(a)
Use the graph to write down the value of \(y\) when \(x = 2\).

(b)
Use the graph to write down the value of \(x\) when \(y = 1\).
3 marks
The graph of \(y = \frac{ 6 }{ x }\) is drawn on the grid.

Both branches bend towards the axes without ever arriving.

(a)
Calculate the value of \(y\) when \(x = 2000\).

Give your answer as a decimal.

(b)
Explain why the graph never touches the \(x\)-axis.

(c)
Explain why the graph never touches the \(y\)-axis.
5 marks
A curve has equation \(y = \frac{ a }{ x }\), where \(a\) is a number.

The curve passes through the point \((2, 6)\).

(a)
Work out the value of \(a\).

(b)
Work out the value of \(y\) when \(x = 6\).

(c)
Work out the value of \(x\) when \(y = -3\).
2 marks
The graph of \(y = x^3 - 12x - 2\) is drawn on the grid, together with the lines \(y = 14\) and \(y = -10\).

On the \(x\)-axis each square is \(1\). On the \(y\)-axis the numbered lines are \(4\) apart, and the fainter lines between them are \(1\) apart.

(a)
Write down the number of solutions of \(x^3 - 12x - 2 = 14\)

(b)
Write down the number of solutions of \(x^3 - 12x - 2 = -10\)
4 marks
The graph of \(y = -x^3 + 12x + 2\) is drawn on the grid.

On the \(x\)-axis each square is \(1\). On the \(y\)-axis the numbered lines are \(4\) apart, and the fainter lines between them are \(1\) apart.

(a)
Somewhere on the grid the curve stops rising and starts falling.

Write down the coordinates of that peak.

(b)
Somewhere else on the grid the curve stops falling and starts rising.

Write down the coordinates of that low point.
3 marks The table shows some values of \(y = x^3 - 5x - 2\)

x -3 -2 -1 0 1 2 3

y 0 -6 -4 10

Three of the \(y\) values are missing.
(a)
Calculate the value of \(y\) when \(x = -3\).

(b)
Calculate the value of \(y\) when \(x = -1\).

(c)
Calculate the value of \(y\) when \(x = 0\).

x	-3	-2	-1	0	1	2	3
y	-14		6	4		6

x	-3	-2	-1	0	1	2	3
y		0			-6	-4	10

Answers

(a) \(2\)
(b) \(2\)
(c) \(22\)
(a) \(4\)
(b) \(8\)
(a) \(0.003\)
(b) Because y is only 0 when the numerator is 0, and it never is — dividing a fixed number by x always leaves something, however large x gets.
(c) Because a point on the y-axis has x = 0, and you cannot divide by 0, so there is no value of y there.
(a) \(12\)
(b) \(2\)
(c) \(-4\)
(a) \(2\)
(b) \(3\)
(a) \((2, 18)\)
(b) \((-2, -14)\)
(a) \(-14\)
(b) \(2\)
(c) \(-2\)

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