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Cubic/reciprocal graphs

Free Printable Cubic/reciprocal graphs PDF Worksheets, Revision Guide & GCSE Past Paper Questions (Grade 5)

Download free cubic/reciprocal graphs worksheets (PDF), GCSE maths past paper questions and worked solutions for Cubic/reciprocal graphs (5K). This is a Grade 5 algebra topic from the GCSE questions by topic collection for Foundation and Higher tier crossover GCSE Maths (Edexcel, AQA, OCR and IGCSE).

The free PDF worksheets table below brings together printable question booklets (Qs) and worked answers or mark schemes (Ans) from IGCSE MT, Maths Genie, Maths4Everyone, Corbett Maths, Maths Made Easy, Dr Austin, GCSE Maths Rocket, E Classroom and PMT, along with past paper question booklets.

Below the worksheet links you can read our cubic/reciprocal graphs revision notes and worked examples with diagrams, try real GCSE past paper cubic/reciprocal graphs questions with step-by-step solutions, or download a PDF of all cubic/reciprocal graphs worksheet links.

Free Cubic/reciprocal graphs PDF Worksheets & Answers

Grade 5 • Topic 5K
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GCSE past paper questions & solutions
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IGCSE (9–1) past paper booklet
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Graded exam questions & solutions
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GCSE 9–1 practice booklet & solutions
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MME worksheet & model answers
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PMT (Edxl)
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PMT (IGCSE)
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Grade 5 topics
5A - Combining ratios5B - Direct and inverse proportion
5C - Reverse percentages5D - Standard form
5E - Speed5F - Changing the subject
5G - Quadratic expressions5H - Solving quadratic equations
5J - Drawing quadratic graphs5K - Cubic/reciprocal graphs
5L - Simultaneous equations5M - Sector areas and arc lengths
5N - Sim. equ. graphically5P - Midpoint of a line segment
5Q - Gradient of a line5R - Equation of a line
5S - Spheres and cones5T - Similar shapes (lengths)
5U - Trigonometry (SOHCAHTOA)5V - Exact trig. values
5W - Probability trees5X - Vectors
5Y - Venn diagrams5Z - Density
5AA - Pressure
GCSE Maths Grade 5 Revision Notes & Method

Guide to Cubic/reciprocal graphs: Explanation, Diagrams & Worked Examples

Download Printable One Page Guide (PDF)

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−2−1012
y−410−14
x = −2: (−2)3 − 2(−2) = −4
x = 2: 23 − 2(2) = 4
x y

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

Solution:

x−6−3−2236
y−1−2−3321
Curves never touch axes:
Asymptotes: x = 0, y = 0
x y
Scanned GCSE Exam Questions & Worked Solutions

GCSE Past Paper Cubic/reciprocal graphs Questions & Answers

Past Paper Example 1 AQA Jun 2023 1F Q23 / 1H Q9 2 marks
Full Paper (PDF) • Mark Scheme (PDF)
Erika tries to sketch the graph y = 1x with x ≠ 0
xyO
Make **two** different criticisms of her sketch.
Worked Solution & Mark Scheme (2 marks)
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
Past Paper Example 2 GCSE Nov 2019 2H Q5 2 marks
Full Paper (PDF) • Mark Scheme (PDF)
Here are six graphs.
AxyOBxyOCxyODxyOExyOFxyO
(a) y = x3 [1 mark]
(b) y = 1x [1 mark]
Worked Solution & Mark Scheme (2 marks)
Part (a) Solution:
y = x³ passes through (0, 0) with positive cubic shape (graph F) B1
Answer (a): F
Part (b) Solution:
y = 1/x has branches in the 1st and 3rd quadrants (graph D) B1
Answer (b): D
Past Paper Example 3 GCSE Nov 2022 1H Q9 4 marks
Full Paper (PDF) • Mark Scheme (PDF)
(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]
xy–39–2–101243–9
(b) On the grid, draw the graph of y = 6x - x3 for values of x from –3 to 3 [2 marks]
xy–3–2–1123–10–8–6–4–2246810O
Worked Solution & Mark Scheme (4 marks)
Part (a) Solution:
x = –2: –12 – (–8) = –4 M1
x = –1: –6 – (–1) = –5
x = 0: 0
x = 1: 6 – 1 = 5 A1
xy–39–2–4–1–50015243–9
Answer (a): –4, –5, 0, 5
Part (b) Solution:
x = –2: –12 – (–8) = –4 M1
x = –1: –6 – (–1) = –5
x = 0: 0
x = 1: 6 – 1 = 5 A1
xy–3–2–1123–10–8–6–4–2246810O
Answer (b): Smooth cubic curve through (–3, 9), (–2, –4), (–1, –5), (0, 0), (1, 5), (2, 4), (3, –9)

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