Cubic/reciprocal graphs Quiz (Grade 5)

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Cubic/reciprocal graphs — Notes & Worked Examples Show / Hide

Explanation:

A cubic graph (y = ax³ + bx² + cx + d) forms a smooth S-shaped wave that goes from bottom-left to top-right when a > 0 (or top-left to bottom-right when a < 0). Remember that cubing a negative number gives a negative result, e.g. (−2)³ = −8.

A reciprocal graph (y = kx, x ≠ 0) has two separate curves in opposite quadrants (1st and 3rd if k > 0). It never touches the x-axis or y-axis; the axes x = 0 and y = 0 are called asymptotes.

Example 1: Complete the table of values for the cubic graph y = x³ − 2x for −2 ≤ x ≤ 2.

Solution:

x	−2	−1	0	1	2
y	−4	1	0	−1	4

x = −2: (−2)³ − 2(−2) = −4

x = 2: 2³ − 2(2) = 4

Example 2: Complete the table for y = 6x and write down the equations of its asymptotes.

Solution:

x	−6	−3	−2	2	3	6
y	−1	−2	−3	3	2	1

Curves never touch axes:

Asymptotes: x = 0, y = 0

One Page Guide (PDF) Cubic/reciprocal graphs Worksheets

Question 1 OCR Nov 2022 5H Q21

[1 mark]

The graph of y = 1x - 2 is drawn on the grid for -2 ≤ x ≤ 6.

(a) There are no values of x for which 1x - 2 = k.
Find the value of k. [1 mark]

k =

Worked Solution & Answer

(a)

The horizontal asymptote is y = 0, so 1/(x − 2) never equals 0 ⇒ k = 0. B1

Answer (a): k = 0

Question Paper (PDF) • Mark Scheme (PDF)

Question 2 AQA Jun 2023 1F Q23 / 1H Q9

[2 marks]

Erika tries to sketch the graph y = 1x with x ≠ 0

Make **two** different criticisms of her sketch.

Worked Solution & Answer

1. For negative values of x, y = 1/x is negative, so the left branch should be in the bottom-left quadrant. M1
2. The y-axis (x = 0) is an asymptote, so the graph must not touch the y-axis. A1

Answer: For x < 0, the curve should be below the x-axis (in the 3rd quadrant)
The curve should not touch or cross the y-axis

Question Paper (PDF) • Mark Scheme (PDF)

Question 3

[2 marks]

Here are six graphs.

(a) y = x³ [1 mark]

(b) y = 1x [1 mark]

Worked Solution & Answer

(a)

y = x³ passes through (0, 0) with positive cubic shape (graph F) B1

Answer (a): F

(b)

y = 1/x has branches in the 1st and 3rd quadrants (graph D) B1

Answer (b): D

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Question 4 AQA Nov 2019 3H Q14

[3 marks]

A graph has equation y = x³ + a where a is an integer.
The graph passes through the point (3, 29).

Draw the graph for values of x from -3 to 3.

Worked Solution & Answer

Find a: substitute (3, 29): M1
29 = 3³ + a ⇒ 29 = 27 + a ⇒ a = 2
So y = x³ + 2
Table of points:
x = -3 ⇒ y = -27 + 2 = -25
x = -2 ⇒ y = -8 + 2 = -6 M1
x = -1 ⇒ y = -1 + 2 = 1
x = 0 ⇒ y = 2
x = 1 ⇒ y = 3
x = 2 ⇒ y = 10
x = 3 ⇒ y = 29
Plotted with a smooth cubic curve. A1

Answer: Value of a a = 2

Question Paper (PDF) • Mark Scheme (PDF)

Question 5 IGCSE Jun 2025 1H Q10

[3 marks]

(a) Complete the table of values for y = x³ + 2x + 3. (1) (b) On the grid, draw the graph of y = x³ + 2x + 3 for –3 ≤ x ≤ 3. (2)

(a) Complete the table of values for y = x³ + 2x + 3 [1 mark]

(b) On the grid, draw the graph of y = x³ + 2x + 3 for -3 ≤ x ≤ 3 [2 marks]

Worked Solution & Answer

(a)

x = –3 ⇒ (–3)³ + 2(–3) + 3 = –27 – 6 + 3 = –30 B1

Answer (a): –30, (–9), (0), (3), (6), 15, (36)

(b)

x = 2 M1
⇒ 2³ + 2(2) + 3 = 8 + 4 + 3 = 15 A1

Answer (b): Smooth cubic curve through (–3, –30), (–2, –9), (–1, 0), (0, 3), (1, 6), (2, 15), (3, 36)

Question Paper (PDF) • Mark Scheme (PDF)

Question 6

[3 marks]

Here are some graphs.

Write down the letter of the graph that could have the equation

(i) y = x² - 4 [1 mark]

(ii) y = -x³ [1 mark]

(iii) y = -5x [1 mark]

Worked Solution & Answer

(i)

Positive parabola with y-intercept –4 is H B1

Answer (i): H

(ii)

Decreasing cubic through origin is F B1

Answer (ii): F

(iii)

Reciprocal in 2nd and 4th quadrants is J B1

Answer (iii): J

Question Paper (PDF) • Mark Scheme (PDF)

Question 7

[4 marks]

(a) Complete the table of values for y = 6x – x³. (2) (b) Draw the graph of y = 6x – x³ for x from –3 to 3. (2)

(a) Complete the table of values for y = 6x - x³ [2 marks]

(b) On the grid, draw the graph of y = 6x - x³ for values of x from –3 to 3 [2 marks]

Worked Solution & Answer

(a)

x = –2: –12 – (–8) = –4 M1
x = –1: –6 – (–1) = –5
x = 0: 0
x = 1: 6 – 1 = 5 A1

Answer (a): –4, –5, 0, 5

(b)

x = –2: –12 – (–8) = –4 M1
x = –1: –6 – (–1) = –5
x = 0: 0
x = 1: 6 – 1 = 5 A1

Answer (b): Smooth cubic curve through (–3, 9), (–2, –4), (–1, –5), (0, 0), (1, 5), (2, 4), (3, –9)

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Question 8 AQA Jun 2019 1F Q26 / 1H Q10

[4 marks]

The number of items, n, made in 1 hour by a machine is given by n = 60t
t is the time in minutes the machine takes to make one item.
The value of t changes for different types of item.

(a) On the grid below, draw the graph of n = 60t for values of t from 1 to 4 [2 marks]

(b) The machine takes 3 minutes 30 seconds to make one item.
Use your graph to estimate the value of n. [2 marks]

Answer

Worked Solution & Answer

(a)

Calculate coordinates: M1
When t = 1, n = 60
When t = 2, n = 30
When t = 3, n = 20
When t = 4, n = 15
Plot these points and join with a smooth curve. A1

Answer (a): Answer Smooth reciprocal curve through (1, 60), (2, 30), (3, 20), (4, 15)

(b)

3 minutes 30 seconds = 3.5 minutes M1
From t = 3.5 on the graph, read off n ≈ 17 (or 16 to 18).
603.5 = 17.14 A1

Answer (b): Answer 17

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Question 9

[4 marks]

(a) Complete the table of values for y = 6/x. (2) (b) Draw the graph of y = 6/x for 0.5 ⩽ x ⩽ 6. (2)

(a) Complete the table of values for y = 6x [2 marks]

(b) On the grid below, draw the graph of y = 6x for values of x from 0.5 to 6 [2 marks]

Worked Solution & Answer

(a)

6/0.5 = 12 M1
6/1.5 = 4
6/3 = 2
6/5 = 1.2
6/6 = 1 A1

Answer (a): 12, (6), 4, (3), 2, (1.5), 1.2, 1

(b)

Smooth curve through (0.5, 12), (1, 6), (1.5, 4), (2, 3), (3, 2), (4, 1.5), (5, 1.2), (6, 1) M1 A1

Answer (b): Smooth curve through (0.5, 12), (1, 6), (1.5, 4), (2, 3), (3, 2), (4, 1.5), (5, 1.2), (6, 1)

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Question 10 IGCSE Nov 2023 1H Q12

[5 marks]

(a) Table of values for y = x³ − 3x + 1 (2) (b) Draw graph of y = x³ − 3x + 1 (2) (c) Letter of graph with equation y = −1/x² (1)

(a) Complete the table of values for y = x³ - 3x + 1 [2 marks]

(b) On the grid, draw the graph of y = x³ - 3x + 1 for values of x from -2 to 2 [2 marks]

Worked Solution & Answer

(a)

(−2)³ − 3(−2) + 1 = −1 M1
(−1)³ − 3(−1) + 1 = 3
0 + 1 = 1
2³ − 6 + 1 = 3 A1

Answer (a): y = −1, 3, 1, (−1), 3

(b)

Smooth cubic curve through (−2, −1), (−1, 3), (0, 1), (1, −1), (2, 3) M1 A1

Answer (b): Smooth cubic curve through (−2, −1), (−1, 3), (0, 1), (1, −1), (2, 3)

(c)

y = −1/x² is always negative for all x ≠ 0 ⇒ Graph D B1

Answer (c): D

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Score: 0 / 10 (0 / 31 marks)

Cubic/reciprocal graphs Worksheets Next: Simultaneous equations →

Grade 5 Quizzes

25 Topics

5A - Combining ratios	5B - Direct and inverse proportion
5C - Reverse percentages	5D - Standard form
5E - Speed	5F - Changing the subject
5G - Quadratic expressions	5H - Solving quadratic equations
5J - Drawing quadratic graphs	5K - Cubic/reciprocal graphs
5L - Simultaneous equations	5M - Sector areas and arc lengths
5N - Sim. equ. graphically	5P - Midpoint of a line segment
5Q - Gradient of a line	5R - Equation of a line
5S - Spheres and cones	5T - Similar shapes (lengths)
5U - Trigonometry (SOHCAHTOA)	5V - Exact trig. values
5W - Probability trees	5X - Vectors
5Y - Venn diagrams	5Z - Density
5AA - Pressure