IGCSE Maths / Graphs and transformations

Drawing cubic graphs revision notes

Edexcel International GCSE Maths (4MA1), chapter 26: Graphs and transformations, specification point 3.3a. Free lesson.

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A cubic graph comes from an equation with an x³ term and has a smooth, wavy shape. Make a table of values, plot every point and use the sign of the x³ term to check which way the graph runs.

How to make a table of values

Substitute each given x value into the equation to find y. Work out the cube first, keeping its sign, then do the multiplication and subtraction.

Worked table for y = x³ − 3x

For each x value, calculate x³ − 3x. The negative inputs are especially important: cubing a negative gives a negative result.

How to draw the cubic

Plot each coordinate from the table. The graph of y = x³ − 3x climbs through (−2, −2), reaches a turning point at (−1, 2), then falls through the origin and reaches another turning point at (1, −2) before climbing again. A cubic has a hump and a dip, so it can have two turning points. The curve crosses the x-axis at (0, 0) in this example.

What the sign of x³ tells you

Check the sign in front of the cubed term before drawing. A positive x³ term makes the graph climb from bottom left to top right. A negative x³ term turns that overall direction over, so the graph falls from top left to bottom right. For y = 4x − x³, the graph crosses the x-axis at (−2, 0), (0, 0) and (2, 0). A cubic can have up to three x-axis crossings.

Check negative cubes

A common error is writing a negative number cubed as positive. For example, (−2)³ = (−2) × (−2) × (−2) = 4 × (−2) = −8. Squaring a negative gives a positive, but cubing it leaves a negative. Quick check: for y = x³ − 2x when x = −3, y = (−3)³ − 2 × (−3) = −27 + 6 = −21.