Worksheets · Foundation and Higher

Cubic, reciprocal and contextual graphs

8 exam-style questions, grades 3 to 4. Worked solutions and the marks are on the last page.

Question 1Non-calculator · 2 marks
Here are four equations: $y = 2x$ , $y = x^2$ , $y = x^3$ and $y = \dfrac{1}{x}$ .
(a) Which equation has a graph that never touches either axis? (1)
1. $y = 2x$
2. $y = x^2$
3. $y = x^3$
4. $y = \frac{1}{x}$
(b) Which equation has a graph passing through $(2, 8)$ and $(-2, -8)$ ? (1)
1. $y = 2x$
2. $y = x^2$
3. $y = x^3$
4. $y = \frac{1}{x}$
Question 2Non-calculator · 3 marks
The time, $t$ hours, to empty a pool with $n$ identical pumps is given by $t = \dfrac{60}{n}$ .
(a) Work out the time taken with 4 pumps. (1)
(b) How many pumps are needed to empty the pool in 5 hours? (2)
Question 3Calculator · 2 marks
(a) Point P lies on the graph y = 18/x. The x-coordinate of P is $-6.$ Work out its y-coordinate. (2)
Question 4Non-calculator · 4 marks
A fixed rectangle has area 24 cm². Its positive side lengths x and y satisfy $xy=24$ .
(a) Write y in terms of x and find y when x = 6. (3)
(b) Describe what happens to y when x is doubled. (1)
Question 5Non-calculator · 4 marks
The curve has equation $y=x^3-3$ .
(a) Find the y-coordinate when x = −2. (2)
(b) Find the change in y as x increases from 1 to 3. (2)
Question 6Non-calculator · 4 marks
A journey’s distance-time graph joins $(0,0)$ , $(4,48)$ , $(7,48)$ and $(10,84)$ by straight segments. Time is in minutes and distance in metres.
(a) Find the average speed over the whole journey. (2)
(b) Find the length of the stop. (2)
Question 7Non-calculator · 3 marks
A graph of the temperature of a cooling drink is a smooth curve through (0, 96), (10, 64), (20, 48) and (30, 40). Time is in minutes and temperature in °C.
(a) Find the average rate of cooling over the first 10 minutes, in °C per minute. (2)
(b) Describe how the rate of cooling changes over the 30 minutes. Use the data. (1)
Question 8Non-calculator · 4 marks
The graph of $y = \frac{12}{x}$ is drawn for $x > 0$ .
(a) Find y when x = 0.5. (2)
(b) The line y = 3x crosses the curve. Find the coordinates of the crossing point. (2)

Worked solutions and marks

Question 1

(a) $y = \frac{1}{x}$

$\frac{1}{x}$ is never 0, and $x = 0$ is not allowed, so the reciprocal graph never meets either axis.

B1 $y = \frac{1}{x}$ .

(b) $y = x^3$

$2^3 = 8$ and $(-2)^3 = -8$ .

B1 The correct answer, $y = x^3$ .

Question 2

(a) $15$ hours

$t = 60 \div 4 = 15$ hours.

B1 The correct answer, $15$ hours.

(b) $12$

$5 = \frac{60}{n}$ , so $n = 60 \div 5 = 12$ .

M1 Writing $5 = \frac{60}{n}$ or $n = \frac{60}{5}$ .
A1 The correct answer, $12$ .

Question 3

(a) $-3$

$18/(-6)$
Substitute x = $-6$ into the reciprocal rule.
y = 18/( $-6$ ) = $-3.$

P1 Establishing $18/(-6)$ or an equivalent valid method.
A1 Correct answer: $-3$

Question 4

(a) $4$ cm

Divide the area by x.
$y=24/x$
Substitute x = 6.
$24/6$
Therefore $4$ cm.

P1 Divide the area by x.
P1 Substitute x = 6.
A1 Correct answer: $4$ cm

(b) y is halved because the product xy remains equal to the fixed area.

Therefore y is halved because the product xy remains equal to the fixed area.

C1 Correct conclusion with supporting reasoning: y is halved because the product xy remains equal to the fixed area.

Question 5

(a) $-11$

Cube the signed input before subtracting the constant.
$(-2)^{3}-3$
Therefore $-11$ .

M1 Cube the signed input before subtracting the constant.
A1 Correct answer: $-11$

(b) $26$

Evaluate both outputs before subtracting.
$(3^{3}-3)-(1^{3}-3)$
Therefore $26$ .

M1 Evaluate both outputs before subtracting.
A1 Correct answer: $26$

Question 6

(a) $8.4$ m/min

Use total distance divided by total elapsed time, including the stop.
$84/10$
Therefore $8.4$ m/min.

P1 Use total distance divided by total elapsed time, including the stop.
A1 Correct answer: $8.4$ m/min

(b) $3$ minutes

Subtract the times at the ends of the horizontal segment.
$7-4$
Therefore $3$ minutes.

M1 Subtract the times at the ends of the horizontal segment.
A1 Correct answer: $3$ minutes

Question 7

(a) $3.2$ °C per minute

Divide the fall in temperature by the time taken.
$\frac{96-64}{10}$
Therefore $3.2$ °C per minute.

M1 Divide the fall in temperature by the time taken.
A1 Correct answer: $3.2$ °C per minute

(b) The drink cools more slowly as time passes: it falls 32 °C in the first 10 minutes, 16 °C in the next 10 and 8 °C in the last 10.

The drink cools more slowly as time passes: it falls 32 °C in the first 10 minutes, 16 °C in the next 10 and 8 °C in the last 10.

C1 Correct conclusion with supporting reasoning: The drink cools more slowly as time passes: it falls 32 °C in the first 10 minutes, 16 °C in the next 10 and 8 °C in the last 10.

Question 8

(a) $24$

Divide 12 by 0.5.
$12/0.5$
Therefore $24$ .

M1 Divide 12 by 0.5.
A1 Correct answer: $24$

(b) $(2,6)$

Set 3x equal to 12/x and solve.
$3x^{2}=12$
Therefore $(2,6)$ .

M1 Set 3x equal to 12/x and solve.
A1 Correct answer: $(2,6)$

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Cubic, reciprocal and contextual graphs

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