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OCR MathsAlgebra

Cubic, reciprocal and exponential graphs

Recognise and interpret common graph shapes.

OCRGCSE MathsAlgebraHigher
Visual model

Recognise common Higher graph shapes

cubicexponential
Gold-standard guide
26 mins

What you will learn

Recognise and interpret common graph shapes.
Use a clear step-by-step method for cubic, reciprocal and exponential graphs.
Check your answer and avoid the most common exam mistake.
Useful before you start
Core number skillsEarlier algebra skillsShowing clear working
Core knowledge

Know the rule, then use it

These are the short notes. Read each one, then check you can use it in the worked example below.

Method

Method

Recognise graph families by shape: cubics have an S-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1)

Step 1

Factorise to find x-intercepts

x3 − 4x = x(x2 − 4) = x(x − 2)(x + 2)

Step 2

Find the y-intercept

When x = 0: y = 0

Step 3

Determine the shape using the positive leading coefficient

Because the x3 coefficient is positive, the graph rises from bottom-left to top-right with an S-bend through the three roots

Watch out

Watch out

Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve

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Core method

Recognise graph families by shape: cubics have an S-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1)

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Exam check

Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve

Worked example

Sketch y = x3 − 4x and state the values of x where it crosses the x-axis.

1

Factorise to find x-intercepts: x3 − 4x = x(x2 − 4) = x(x − 2)(x + 2). Roots at x = 0, x = 2 and x = −2.

2

Find the y-intercept: When x = 0: y = 0. The graph passes through the origin.

3

Determine the shape using the positive leading coefficient: Because the x3 coefficient is positive, the graph rises from bottom-left to top-right with an S-bend through the three roots.

4

Sketch, marking all intercepts: Mark (−2, 0), (0, 0) and (2, 0) and draw a smooth S-shaped curve.

Final answer

x-intercepts at x = −2, 0 and 2; y-intercept at 0

Question ladder

Build up to the hardest questions

Do them in order. If you miss a step, read the solution, then redo the question without looking.

Workedreasoning

Sketch y = x3 − 4x and state the values of x where it crosses the x-axis.

4 marks4 minscubic-reciprocal-and-exponential-graphs-worked
Show solution
Worked solution
  1. 1.Factorise to find x-intercepts: x3 − 4x = x(x2 − 4) = x(x − 2)(x + 2). Roots at x = 0, x = 2 and x = −2.
  2. 2.Find the y-intercept: When x = 0: y = 0. The graph passes through the origin.
  3. 3.Determine the shape using the positive leading coefficient: Because the x3 coefficient is positive, the graph rises from bottom-left to top-right with an S-bend through the three roots.
  4. 4.Sketch, marking all intercepts: Mark (−2, 0), (0, 0) and (2, 0) and draw a smooth S-shaped curve.
Final answer

x-intercepts at x = −2, 0 and 2; y-intercept at 0

Mark points
  • M1: factorise to find x-intercepts
  • M1: find the y-intercept
  • M1: determine the shape using the positive leading coefficient
  • M1: sketch, marking all intercepts
  • A1: x-intercepts at x = −2, 0 and 2; y-intercept at 0
Watch out

Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve.Also, y = 1/x is not defined at x = 0 — never draw it crossing the axes.

Diagnosticrecall

Sketch y = x3. Where does it cross the axes?

1 mark2 minscubic-reciprocal-and-exponential-graphs-q1
Show solution
Worked solution
  1. 1.Spot the skill: Recognise graph families by shape: cubics have an S-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  2. 2.Use the factorise to find x-intercepts stage first, then find the y-intercept.
  3. 3.Keep the final answer visible: Only at the origin (0, 0).
Final answer

Only at the origin (0, 0)

Mark points
  • M1: use the correct recognise graph families by shape: cubics have an s-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  • A1: Only at the origin (0, 0)
Watch out

Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve.Also, y = 1/x is not defined at x = 0 — never draw it crossing the axes.

Easyprocedure

Describe the key features of y = 2/x.

2 marks3 minscubic-reciprocal-and-exponential-graphs-q2
Show solution
Worked solution
  1. 1.Spot the skill: Recognise graph families by shape: cubics have an S-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  2. 2.Use the find the y-intercept stage first, then determine the shape using the positive leading coefficient.
  3. 3.Keep the final answer visible: Two branches in quadrants 1 and 3, never touches the axes, approaches axes as asymptotes.
Final answer

Two branches in quadrants 1 and 3, never touches the axes, approaches axes as asymptotes

Mark points
  • M1: use the correct recognise graph families by shape: cubics have an s-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  • A1: Two branches in quadrants 1 and 3, never touches the axes, approaches axes as asymptotes
Watch out

Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve.Also, y = 1/x is not defined at x = 0 — never draw it crossing the axes.

Mediumreasoning

For y = 3x, state the y-intercept and whether it is increasing or decreasing.

3 marks4 minscubic-reciprocal-and-exponential-graphs-q3
Show solution
Worked solution
  1. 1.Spot the skill: Recognise graph families by shape: cubics have an S-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  2. 2.Use the determine the shape using the positive leading coefficient stage first, then sketch, marking all intercepts.
  3. 3.Keep the final answer visible: y-intercept (0, 1); increasing as x increases.
Final answer

y-intercept (0, 1); increasing as x increases

Mark points
  • M1: use the correct recognise graph families by shape: cubics have an s-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  • A1: y-intercept (0, 1); increasing as x increases
Watch out

Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve.Also, y = 1/x is not defined at x = 0 — never draw it crossing the axes.

Hardproblem solving

Find the x-intercepts of y = x3 − 9x.

3 marks5 minscubic-reciprocal-and-exponential-graphs-q4
Show solution
Worked solution
  1. 1.Spot the skill: Recognise graph families by shape: cubics have an S-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  2. 2.Use the sketch, marking all intercepts stage first, then factorise to find x-intercepts.
  3. 3.Keep the final answer visible: x = −3, 0 and 3.
Final answer

x = −3, 0 and 3

Mark points
  • M1: use the correct recognise graph families by shape: cubics have an s-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  • A1: x = −3, 0 and 3
Watch out

Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve.Also, y = 1/x is not defined at x = 0 — never draw it crossing the axes.

Exam-stylemulti-step

Which graph passes through (1, 3) and (−1, −3): y = 3x2 or y = 3/x?

4 marks6 minscubic-reciprocal-and-exponential-graphs-q5
Show solution
Worked solution
  1. 1.Spot the skill: Recognise graph families by shape: cubics have an S-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  2. 2.Use the factorise to find x-intercepts stage first, then find the y-intercept.
  3. 3.Keep the final answer visible: y = 3/x (since 3/(−1) = −3, satisfying the negative point).
Final answer

y = 3/x (since 3/(−1) = −3, satisfying the negative point)

Mark points
  • M1: use the correct recognise graph families by shape: cubics have an s-curve and can cross the x-axis up to 3 times, reciprocals y = k/x form two curves in opposite quadrants, exponentials y = ax are always above the x-axis and pass through (0, 1).
  • A1: y = 3/x (since 3/(−1) = −3, satisfying the negative point)
Watch out

Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve.Also, y = 1/x is not defined at x = 0 — never draw it crossing the axes.

Grade 9 stretchproblem solving

The curve y = 2x is translated 3 units upwards. State the new equation and its horizontal asymptote.

4 marks7 minsgraphs-shape-g9
Show solution
Worked solution
  1. 1.Add 3 to the function output.
  2. 2.Translate the original asymptote y = 0 upwards by 3.
Final answer

y = 2x + 3, with asymptote y = 3

Mark points
  • M1: add 3 outside the power
  • A1: y = 2x + 3
  • A1: asymptote y = 3
Watch out

Do not rush straight into arithmetic. Select the relevant method and show a complete chain of working.

Timed checkpoint
16 mins - 9 marks

Switch between skills

Set a timer and attempt all four questions before opening any answers. This is closer to the way skills appear in a real paper.

1Cubic, reciprocal and exponential graphs - 2 marksSketch y = x3. Where does it cross the axes?Mark answer
Answer

Only at the origin (0, 0)

2Algebraic notation and substitution - 2 marksFind t² + 4t when t = −2Mark answer
Answer

−4

3Simplifying expressions - 2 marksSimplify 2xy + 3x²y − xy + x²yMark answer
Answer

3x²y + xy

4Expanding and factorising - 3 marksFactorise fully 6x² + 9xMark answer
Answer

3x(2x + 3)

Mastery check
  • I can explain the method for cubic, reciprocal and exponential graphs.
  • I can show clear working without skipping key steps.
  • I can avoid this mistake: Students confuse cubic and quadratic shapes, drawing a U-shape instead of an S-curve.Also, y = 1/x is not defined at x = 0 — never draw it crossing the axes.
Ask Nova Bot
Explain cubic, reciprocal and exponential graphs from the beginning.Give me a hint for a cubic, reciprocal and exponential graphs question.Quiz me on cubic, reciprocal and exponential graphs.
Related topics
Algebraic notation and substitutionSimplifying expressionsExpanding and factorising
Official exam-board sources

This guide follows the OCR GCSE Mathematics J560 specification. Practice questions are original Learnova questions shaped around official content and exam skills.

Official specification Official past papers
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