Step-by-step worked examples and graded practice questions on cubic and reciprocal graphs — plotting from a table of values, finding roots, and identifying asymptotes.
📚 Foundation & Higher✅ 15 Practice Questions🔍 Full Worked Examples⚠️ Common Mistakes
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Cubic and reciprocal graphs are two more families of non-linear graphs you'll meet at GCSE, alongside quadratics. They have very different shapes, but both come up regularly in exam questions on plotting, sketching and finding roots.
Cubic graphs
A cubic graph is the graph of an equation containing an x³ term. Cubic graphs have a distinctive S-shaped curve. If the coefficient of x³ is positive, the curve generally rises from bottom-left to top-right; if negative, it falls from top-left to bottom-right.
The graph of y = x³ − 4x, showing its S-shape and three roots.
Worked Example 1 — Table of values for a cubic graph
Complete a table of values for y = x³ − 4x, for x from −3 to 3, and use it to plot the graph.
1
Substitute each x-value: x = −3 gives y = −27 + 12 = −15. x = −2 gives y = −8 + 8 = 0. x = −1 gives y = −1 + 4 = 3.
2
Continuing: x = 0 gives y = 0. x = 1 gives y = 1 − 4 = −3. x = 2 gives y = 8 − 8 = 0. x = 3 gives y = 27 − 12 = 15.
3
Plot each point and join them with a smooth S-shaped curve
The table of values plotted and joined with a smooth S-shaped curve.
Worked Example 2 — Finding roots by factorising
Find the roots of y = x³ − 4x
1
Factorise: x³ − 4x = x(x² − 4)
2
Factorise further using the difference of two squares: x(x² − 4) = x(x − 2)(x + 2)
3
Set each factor to zero: x = 0, x − 2 = 0, x + 2 = 0
Answerx = −2, x = 0, x = 2
Worked Example 3 — A negative cubic
Sketch y = −x³ + 4x. How does its shape compare with y = x³ − 4x from Worked Example 1?
1
This has the same roots as before (x = −2, 0, 2), since −x³ + 4x = −(x³ − 4x)
2
Because the coefficient of x³ is negative, the curve is reflected — it falls from top-left to bottom-right instead of rising
AnswerSame roots (−2, 0, 2), but the S-shape is flipped upside down
y = −x³ + 4x: same roots as y = x³ − 4x, but the S-shape is flipped.
Reciprocal graphs
A reciprocal graph has the form y = a/x. These graphs have two separate curved branches and never touch the x-axis or y-axis — both axes are called asymptotes, lines the curve gets closer and closer to but never reaches. Substituting x = 0 is never allowed, since dividing by zero is undefined.
The graph of y = 4/x: two separate branches, with both axes as asymptotes.
Worked Example 4 — Table of values for a reciprocal graph
Complete a table of values for y = 4/x, for x = −4, −2, −1, 1, 2, 4
1
Substitute each value: x = −4 gives y = −1. x = −2 gives y = −2. x = −1 gives y = −4.
2
Continuing: x = 1 gives y = 4. x = 2 gives y = 2. x = 4 gives y = 1.
3
Plot the points — notice they form two separate branches, one for negative x and one for positive x
A reciprocal graph y = a/x passes through the point (2, 6). Find the value of a, and state the equation of the graph.
1
Substitute the point into y = a/x: 6 = a/2
2
Multiply both sides by 2: a = 12
Answera = 12, so the equation is y = 12/x
The graph of y = 12/x, passing through (2, 6).
Practice questions
Work through each question before checking the answers. Difficulty is shown for each question.
Foundation (Grade 3–5)
Q1For y = x³ − 4x, find the value of y when x = 1Foundation
Q2For y = x³ − 4x, find the value of y when x = −2Foundation
Q3Find the roots of y = x(x − 3)(x + 1)Foundation
Q4For y = 5/x, find the value of y when x = 1Foundation
Q5For y = 5/x, find the value of y when x = −5Foundation
Higher (Grade 5–7)
Q6Factorise y = x³ − 9x, and find its rootsHigher
Q7A reciprocal graph y = a/x passes through (3, 8). Find a.Higher
Q8Using y = 24/x from Q7, find the value of y when x = 4Higher
Q9Sketch y = x³ + 2x². Find the x-values where the curve crosses or touches the x-axis.Higher
Q10State whether y = −2x³ + 5 is increasing or decreasing overall as x increases, and explain why.Higher
Higher — Hard (Grade 8–9)
Q11The graph of y = x³ − 4x crosses the x-axis at three points. State the three x-values.Grade 8–9
Q12A reciprocal graph y = k/x has y = −3 when x = 4. Find k, then find the value of x when y = 12.Grade 8–9
Q13Find the coordinates of the points where y = x³ − 4x crosses the x-axis, other than the origin.Grade 8–9
Q14A cubic graph y = x³ + bx has a root at x = 5 (other than x = 0). Find b, then find the third root.Grade 8–9
Q15Two points on a reciprocal graph y = k/x are (2, 9) and (m, 3). Find k, then find m.Grade 8–9
Answers
Foundation (Q1–Q5)
Q1−3(1 − 4)
Q20(−8 + 8)
Q3x = 0, x = 3, x = −1
Q45
Q5−1
Higher (Q6–Q10)
Q6x(x − 3)(x + 3); roots x = 0, 3, −3
Q7a = 24(a/3 = 8)
Q86(24 ÷ 4)
Q9x = 0 (touches, repeated root) and x = −2(x³ + 2x² = x²(x + 2) = 0)
Q10Decreasing overall — the negative coefficient of x³ means the curve falls from top-left to bottom-right
Higher — Hard (Q11–Q15)
Q11x = −2, x = 0, x = 2
Q12k = −12; x = −1(k = 4×(−3); 12 = −12/x → x = −1)
Q13(−2, 0) and (2, 0)
Q14b = −25; third root x = −5(25 + b = 0; x³ − 25x = x(x−5)(x+5))
Q15k = 18; m = 6(k = 2×9; 3 = 18/m → m = 6)
Common mistakes
These are the errors Alamin sees most frequently with cubic and reciprocal graphs at GCSE. Recognising them now will save you marks in the exam.
Common Mistake 1
Substituting x = 0 into a reciprocal graph
y = a/x is undefined at x = 0 — never include x = 0 in a table of values for a reciprocal graph, and never draw the curve crossing the y-axis.
Common Mistake 2
Joining the two branches of a reciprocal graph together
The positive-x and negative-x branches of y = a/x are completely separate curves — never draw a line connecting them across the y-axis.
Common Mistake 3
Drawing straight lines between plotted points on a cubic graph
A cubic graph is a smooth S-shaped curve, not a series of straight segments — this is especially easy to get wrong near the "flatter" middle section of the S.
Common Mistake 4
Missing a factor when finding cubic roots
Always factorise fully. x³ − 4x = x(x² − 4) is not fully factorised — the second bracket is a difference of two squares and factorises further to (x − 2)(x + 2), giving three roots, not two.
Common Mistake 5
Forgetting that a repeated root means the curve touches, not crosses
If a factor is squared (like x² in x²(x + 2)), the curve touches the x-axis at that root and turns back, rather than crossing straight through it.
Common Mistake 6
Confusing the sign of the leading coefficient with the direction of the curve
A positive x³ coefficient means the cubic rises overall from left to right; a negative coefficient means it falls overall. This is easy to mix up, especially once other terms are added.
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