Cubic graphs

Topic summary

You will by now be very familiar with quadratic and linear graphs, but you will need to be familiar with other types of graph within the maths a-level.

Shape of a cubic graph

\[f(x) = ax^3 + bx^2 + cx + d\]

If \(a > 0\), the graph starts from the bottom left, curves up, and finishes in the top right.

If \(a < 0\), the graph starts from the top left, curves down, and finishes in the bottom right.

Intercepts

Y-intercept: The graph intersects the y-axis at \(f(0) = d\).

X-intercepts (Roots): The x-intercepts are the values of \(x\) where \(f(x) = 0\). A cubic function can have one, two, or three real roots. To find the roots, solve the cubic equation.

Turning Points

A cubic graph can have one or two turning points, depending on its shape. These are points where the graph changes direction.

Symmetry

Cubic functions generally do not have symmetry unless they are of a special form, such as \(f(x) = ax^3\), which has rotational symmetry about the origin.