Recognise the S-shape of y=x3 and the two-branch curve of y=x1, and plot them from a table. It is on Foundation and Higher, aimed at grade 5. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A cubic y=x3 (and shifts of it) is an S-shape through the origin, steepening away from zero. A reciprocal y=x1 is two curves in opposite quadrants, never crossing the axes (those are asymptotes). y=x21 sits above the x-axis in both positive and negative x.
Recognise the shape from a sketch question, plot a few points to draw one, and read intersections as solutions. Reciprocal graphs are the picture of inverse proportion. Do not join the two branches through x=0.
Cubes keep the sign: (−3)3=−27. Reciprocals y=xk never meet the axes and sit in opposite quadrants when k>0. To test a point, substitute — do not guess from a sketch.
Objectives
Find a y-value on y = x^3.
Find a y-value for a negative x, and on a shifted cubic.
Find a y-value on y = k/x and name the vertical asymptote.
Key points
Sketch the shape of y=x3 and y=−x3, and of y=xk.
Complete a table of values for a cubic, including negative x-values.
Complete a table of values for a reciprocal graph, and explain why there is no value at x=0.
Match equations to graph sketches: linear, quadratic, cubic, reciprocal.
Worked examples
Example 1
On the curve y=x^3, work out the value of y when x=2.
2 marks
Substitute x = 2 into y = x^3
Calculate 2 cubed: 2 × 2 × 2 = 8
State the value: y is 8
Answer: 8
Common slips
Slip: Arthur wrote y = 3x The curve cubes x, not multiplying by 3.
Slip: Amber calculated 2 × 3 = 6 Cubing means multiplying three copies of 2.
Slip: Amber squared 2 and wrote 4. A cube has three factors of 2, not two.
Example 2
On the curve y=x^3, work out the value of y when x=-2.
2 marks
Substitute x = -2 into y = x^3
Calculate (-2) cubed: (-2) × (-2) × (-2) = -8
State the value: y is -8
Answer: -8
Common slips
Slip: Boris dropped the minus sign Keep negative input in brackets.
Slip: Beth wrote 8 The product of three negative numbers is negative.
Slip: Beth wrote -6 Cube means three factors, not multiplying by 3.
Example 3
On the curve y=12/x, work out the value of y when x=3.
3 marks
Substitute x = 3 into the reciprocal formula
Calculate 12 ÷ 3 = 4
State the value: y is 4
Answer: 4
Common slips
Slip: Craig wrote 12 × 3 A reciprocal curve divides by x.
Slip: Chloe cubed 3 to get 27 Divide the numerator 12 by the given x-value.
Slip: Craig wrote 36 The operation is division, not multiplication.
Questions and answers
GCSE Maths cubic and reciprocal graphs questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
On the curve y=x^3, work out the value of y when x=3.
Show answer
Answer: 27
Cube 3: 3 × 3 × 3.
Result is 27.
M1 — Cube 3: 3 × 3 × 3.
A1 — Result is 27.
Question 2 Easy2 marks
On the curve y=x^3, work out the value of y when x=1.
Show answer
Answer: 1
Cube 1: 1 × 1 × 1.
Result is 1.
M1 — Cube 1: 1 × 1 × 1.
A1 — Result is 1.
Question 3 Medium2 marks
On the curve y=x^3, work out the value of y when x=-3.
Show answer
Answer: -27
Cube -3: (-3) × (-3) × (-3).
Three negatives give -27.
M1 — Cube -3: (-3) × (-3) × (-3).
A1 — Three negatives give -27.
Question 4 Medium2 marks
On the curve y=x^3, work out the value of y when x=-1.
Show answer
Answer: -1
Cube -1: (-1) × (-1) × (-1).
Result is -1.
M1 — Cube -1: (-1) × (-1) × (-1).
A1 — Result is -1.
Question 5 Hard2 marks
On the curve y=12/x, work out the value of y when x=4.
Show answer
Answer: 3
Divide 12 by 4.
12 / 4 = 3.
M1 — Divide 12 by 4.
A1 — 12 / 4 = 3.
Question 6 Hard2 marks
On the curve y=12/x, work out the value of y when x=6.
Show answer
Answer: 2
Divide 12 by 6.
12 / 6 = 2.
M1 — Divide 12 by 6.
A1 — 12 / 6 = 2.
Question 7 Exam-style2 marks
On the curve y=x^3, work out the value of y when x=-4.
Show answer
Answer: -64
Cube -4: (-4) × (-4) × (-4).
Three negative factors give -64.
M1 — (-4) × (-4) × (-4), or 64 seen
A1 — -64
Common mistakes
Most lost marks on this topic are one slip.
Multiplying by 3 instead of cubing
Cubing means three copies of the number multiplied together
Losing the negative sign when cubing a negative
A negative number cubed is negative
Multiplying instead of dividing on reciprocal graphs
Reciprocal curves divide the numerator by x
Spot the slip
On the curve y=x^3, work out y when x=2. Ada writes 6. What is y?
8
6
4
Show the slip
8 2×2×2 = 8.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
For any table question, check a pattern (odd symmetry for x3 and xk) before plotting — a broken pattern reveals the wrong value.
When matching sketches, write one reason for each choice ('two branches, so reciprocal') — it protects the mark if the sketch is ambiguous.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.