Cubic graphs

Cubic graphs are the graphs of cubic functions, y = ax³ + bx² + cx + d, where a isn't 0. They have a curvy S shape that runs from one corner of the graph to the opposite one. A positive cubic (a > 0) starts at the bottom left and ends at the top right; a negative cubic does the opposite. A cubic graph always crosses the x-axis at least once and at most three times.

The graph of y = x³

The simplest cubic graph is y = x³. Work out a table of values and plot the points:

Here x is the input and y is x³. Negative numbers cubed stay negative, so the left half of the curve is below the x-axis. The curve flattens out at the origin for a moment, then climbs steeply. It has rotational symmetry about the origin.

Positive and negative cubic graphs

The sign of a, the number in front of x³, decides which way the graph goes:

Here x is the input and y = x³ − 3x. Not every cubic has two turning points: y = x³ has none, only a flat point at the origin. But every cubic follows the start and end directions in the table, because for large x the x³ term is bigger than all the others.

Check your understanding.

How to sketch a cubic graph

A sketch doesn't need plotted points, but it must show the right shape and mark where the graph crosses the axes. From factorised form:

1. Find the roots: set each bracket to zero. These are where the graph meets the x-axis.
2. Find the y-intercept: put x = 0.
3. Decide the shape: positive or negative cubic, from the sign of the x³ term.
4. Check for repeated roots: a squared bracket means the graph touches the x-axis and turns there instead of crossing.
5. Draw a smooth curve through the intercepts in the right direction, and label them.

Example 1: three different roots

Sketch y = (x + 2)(x − 1)(x − 4).

• Roots: x = −2, x = 1 and x = 4.
• y-intercept: (0 + 2)(0 − 1)(0 − 4) = 2 × (−1) × (−4) = 8, so (0, 8).
• Multiplying the x terms gives x³, so it is a positive cubic: up from the bottom left.

Here x is the input and y = (x + 2)(x − 1)(x − 4). The curve rises through −2, turns, comes down through the y-intercept at 8 and the root at 1, turns again and rises through 4.

Example 2: a repeated root

Sketch y = (x − 2)²(x + 1). The roots are x = 2 (repeated) and x = −1, and the y-intercept is (−2)² × 1 = 4. It is a positive cubic. At x = −1 the curve crosses the axis; at x = 2 it just touches the axis and turns back up, because a squared factor never changes sign.

Example 3: a negative cubic

Sketch y = −x(x − 3)(x + 1). The roots are 0, 3 and −1, and the y-intercept is 0. The x terms multiply to −x³, so it is a negative cubic: it comes down from the top left, through −1, turns, comes back up through 0, turns again and falls through 3.

Spot the degree of a polynomial from its graph in this question from the Graphing Functions course.

Drawing a cubic graph from a table of values

At GCSE you may be asked to complete a table and plot a cubic. For y = x³ − 4x:

Plot the points and join them with a smooth curve, not straight lines. The curve meets the x-axis three times, at x = −2, x = 0 and x = 2: the solutions of x³ − 4x = 0. To solve x³ − 4x = 5 from the graph, draw the line y = 5 and read off the x values where it crosses the curve.

Turning points of a cubic (A-level)

At A-level, differentiate to find the turning points exactly. Take y = x³ − 7x² + 10x + 4:

Set this to zero and use the quadratic formula: x = (14 ± √76) ÷ 6, so x ≈ 0.88 and x ≈ 3.79. The second derivative, 6x − 14, is negative at x = 0.88 (a maximum) and positive at x = 3.79 (a minimum). Substituting back gives a maximum at about (0.88, 8.06) and a minimum at about (3.79, −4.21).

If the derivative has only one repeated root, the cubic has a stationary point of inflection instead of two turning points. For y = x³ + 3, dy/dx = 3x² is zero only at x = 0, so the curve flattens at (0, 3) and carries on upwards. The point of inflection guide explains these.

Transforming cubic graphs

Cubics such as y = (x − a)³ + b are translations of y = x³:

• y = (x + 2)³ is y = x³ moved 2 units to the left, so its flat point is at (−2, 0).
• y = x³ − 4 is y = x³ moved 4 units down.
• y = (x − 1)³ − 4 is moved 1 right and 4 down: its flat point is at (1, −4), and its y-intercept is (−1)³ − 4 = −5.
• y = −x³ is a reflection of y = x³ in the x-axis.

Common mistakes

• Getting the sign of a root wrong. The bracket (x + 2) gives the root x = −2.
• Drawing the wrong shape. Check the sign of the x³ term before you start.
• Crossing at a repeated root. At a squared factor the graph touches the axis and turns.
• Joining plotted points with straight lines. Draw a smooth curve.
• Forgetting the y-intercept on a sketch. Questions often have a mark for it.
• Thinking a cubic always has two turning points. It can have none, like y = x³.

Practice questions

1. Is y = 5 − 2x³ a positive or a negative cubic?
2. Find the roots and y-intercept of y = (x − 3)(x + 1)(x + 5).
3. Describe how the graph of y = (x − 2)²(x + 4) meets the x-axis.
4. Describe the transformation that maps y = x³ onto y = (x + 3)³ + 1.

Answers: 1) Negative (the x³ coefficient is −2). 2) Roots 3, −1 and −5; y-intercept (−3) × 1 × 5 = −15, so (0, −15). 3) It crosses at x = −4 and touches the axis at x = 2. 4) A translation 3 units left and 1 unit up.

Practise with a short timed paper

Cubic graphs come up in GCSE Maths Higher and in A-level Maths. The daily GCSE Maths Higher and A-level Maths: Algebra & Functions papers are short timed papers of 8 questions, with a new one every day. To plot any cubic with its roots and turning points, see the cubic graph calculator guide.

Read next

Maths · GraphsCubic graph calculator: plot y = ax³ + bx² + cx + dFree cubic graph calculator: plot y = ax³ + bx² + cx + d, read the roots and y-intercept, and find the turning points, with practice questions.