## Cubic Graphs

cubic function is a polynomial of degree three.

e.g. y = x3 + 3x2 − 2x + 5

Cubic graphs can be drawn by finding the x and y intercepts.
Because cubic graphs do not have axes of symmetry the turning points have to be found using calculus.

### Sketching Cubics

Method 1: Factorisation.

If the equation is in the form y = (x − a)(x − b)(x − c) the following method should be used:

 Step 1: Find the x-intercepts by putting y = 0. Step 2: Find the y-intercept by putting x = 0. Step 3: Plot the points above to sketch the cubic curve.

 e.g. Sketch the graph of y = (x − 2)(x + 3)(x − 1) Step 1: Find the x-intercepts by putting y = 0. 0 = (x − 2)(x + 3)(x − 1) x = 2 or -3 or 1 Step 2: Find the y-intercepts by putting x = 0. y = (0 − 2)(0 + 3)(0 − 1) y = -2 x 3 x -1 y = 6 Step 3: Plot the points and sketch the curve. Note: Functions with a repeated factor have a graph which just touches the x-axis. e.g. y = (x − 2)2(x + 1)

Method 2: Transformation

The graph of the basic cubic y = x3 is shown in the diagram.

This basic cubic is moved or transformed as follows:

 y = ax3 The a has the effect of changing the basic cubic in the y- direction. It affects the steepness of the graph. If a is large, the cubic is steeper. If a is small, the cubic is flatter. If a is negative, the cubic is inverted. y = x3 + k The k has the effect of moving the cubic up or down the y-axis by k units. y = (x − h)3 The h has the effect of moving the basic cubic along the x-axis by h units.