Cubic graph calculator

A cubic graph calculator plots y = ax³ + bx² + cx + d and shows its key features: where it crosses the x-axis (up to three roots), where it crosses the y-axis (at d), and its turning points. Our Cubic Graph calculator draws the curve as you type the four coefficients, lists the intercepts, and can mark the turning points and draw the derivative.

It opens on y = x³ + 2x² − x + 1. Type new values for a, b, c and d in the four boxes and the curve redraws.

Try it

Cubic Graph

Plot and analyse cubic functions

Open the calculatorGo to its page

How to use the cubic graph calculator

1. Under Equation, type a, b, c and d in the four boxes. The preview shows the equation.
2. Read the Intercepts table for the roots on the x-axis and the y-intercept.
3. Under Analysis, turn on Min / Max to mark the turning points, f′(x) to draw the derivative, or ∫ dx for the area under the curve.
4. Drag and zoom the graph to read off points.

Practice and help sets cubic graph questions with worked solutions.

The shape of a cubic graph

Every cubic has at least one real root, because its two ends go off in opposite directions. Our polynomial guide covers polynomials of every degree.

Worked examples you can check in the calculator

Example 1: a factorised cubic

y = (x − 1)(x − 2)(x − 3) multiplies out to x³ − 6x² + 11x − 6. Its roots are 1, 2 and 3, and its y-intercept is (0, −6). Type 1, −6, 11 and −6 and the intercepts table lists all four points.

Example 2: turning points

For the same cubic, the turning points are where the gradient is zero:

So x = 1.423 and x = 2.577, giving a maximum at (1.423, 0.385) and a minimum at (2.577, −0.385). Turn on Min / Max and the calculator marks both. Turn on f′(x) and you can see the derivative cross zero at the same two x values.

Example 3: a repeated root

y = x²(x − 3) = x³ − 3x² has a repeated root at x = 0, where the curve touches the x-axis and turns, and a single root at x = 3, where it crosses. Type 1, −3, 0 and 0 to see it.

Check yourself:

What the calculator does not do

• Decimal answers, not exact ones. Roots and turning points are found numerically, so 2 + 1/√3 shows as 2.577. Work out exact values by hand.
• No factorising. To factorise a cubic, use the factor theorem or the synthetic division calculator guide.
• One curve at a time. To find where a cubic meets a line, solve the equations together.

Common mistakes

• Forgetting a cubic can have only one real root. Not every cubic crosses the x-axis three times.
• Mixing up a repeated root and a turning point. At a repeated root the curve touches the axis and turns; elsewhere a turning point need not be on the axis.
• Getting the end behaviour backwards. Check the sign of a, the x³ coefficient.
• Dropping zero coefficients. For x³ − 3x², type 0 for c and d.

Practise graphs in a short timed paper

Cubic graphs and their turning points come up in AP Precalculus and AP Calculus AB practice, and in A-level Maths: Algebra & Functions in the UK. Each daily paper is a short timed paper of 8 questions worth 25 to 35 marks, and the papers mix topics. The Graphing Functions course has a polynomial shape explorer.

Read next

Maths · GraphsCubic graphs: shapes, roots and how to sketch themWhat cubic graphs look like, positive and negative cubics, how to sketch one from its roots and y-intercept, repeated roots and turning points.