20.1 Graphs of Cubic and Reciprocal Functions

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: tables of values, plotting and reading cubic and reciprocal graphs. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Complete a table of values for a cubic function such as y = x³ − 3x and draw its graph
• Use a cubic graph to estimate values and solutions
• Draw and recognise the graph of a reciprocal function y = k/x
• Recognise the shapes of linear, quadratic, cubic and reciprocal graphs

Key words

• Cubic: A function whose highest power of x is 3, such as y = x³ or y = x³ − 3x + 1.
• Reciprocal: A function of the form y = k/x. The reciprocal of x is 1/x.
• Table of values: A table of x values and the matching y values, used to plot points.
• Estimate: A value read from a graph. It is not exact, so give it to a sensible accuracy.

Drawing a cubic graph

Substitute each x value into the equation. Take care with negatives: (−2)³ = −8. Plot the points and join them with a smooth curve, not straight lines.

Worked example: table for y = x³ − 3x

When x = −2: y = (−2)³ − 3 × (−2) = −8 + 6 = −2
When x = −1: y = −1 + 3 = 2. When x = 1: y = 1 − 3 = −2. When x = 2: y = 8 − 6 = 2

Answer: see the table

• y; −2 −2; −1 2; 0 0; 1 −2; 2 2

Key fact: A cubic graph with a positive x³ term starts low on the left and ends high on the right. It can have up to two turning points.

Using a cubic graph

To solve an equation such as x³ − 3x = 1, draw the line y = 1 and read off the x values where it crosses the curve.

Worked example: solving from the graph

The line y = 1 crosses the curve three times.
Reading from the graph: x ≈ −1.5, x ≈ −0.3 and x ≈ 1.9

Answer: x ≈ −1.5, −0.3 or 1.9

Reciprocal graphs

For y = k/x you cannot use x = 0, because you cannot divide by 0. The graph has two separate parts. Each part gets closer and closer to the axes but never touches them.

Worked example: table for y = 6/x

x = 1, 2, 3, 6 gives y = 6, 3, 2, 1
x = −1, −2, −3, −6 gives y = −6, −3, −2, −1

Answer: plot both parts as smooth curves

Recognising graph shapes

Look at the highest power of x, or whether x is on the bottom of a fraction.

• Linear; Example y = 2x + 1; Shape straight line
• Quadratic; Example y = x² − 4; Shape U shape (∩ if the x² term is negative), symmetrical
• Cubic; Example y = x³ − 3x; Shape S-shaped curve
• Reciprocal; Example y = 5/x; Shape two separate curves, never touching the axes

Worked example: which graph?

y = 4 − x² has highest power 2 with a negative sign.

Answer: an upside-down U (∩) quadratic

Common mistakes

• Cubing a negative wrongly: (−3)³ = −27, not 27.
• Joining points with straight lines. Cubic and reciprocal graphs are smooth curves.
• Trying to plot y = k/x at x = 0, or joining the two parts of a reciprocal graph across the y-axis.
• Giving only one solution when a line crosses a cubic three times.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out the value of y = x³ - 3x when x = -2.

Answer: -2

Working: (-2)³ - 3 × (-2) = -8 + 6 = -2

Question 2. Work out the value of y = 12/x when x = -4.

Answer: -3

Working: 12 ÷ (-4) = -3

Question 3. What shape is the graph of y = x³ + 1? (Choose from: An S-shaped curve, A U-shaped curve, A straight line, Two separate curves.)

Answer: An S-shaped curve

Working: The highest power is x³, so it is a cubic graph: an S-shaped curve.

Related lessons

• Previous lesson: 19.7 Vectors 2
• Next lesson: 20.2 Non-linear Graphs
• In this chapter: 20.3 Solving Simultaneous Equations Graphically
• In this chapter: 20.4 Solving Simultaneous Equations Algebraically
• In this chapter: 20.5 Rearranging Formulae

Continue in the full course

This lesson is section 20.1 (EGCMF2001) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.