A cubic function will have an x3 term. An example of a cubic function is 2x3 + 8x2 - 2x - 8.
A cubic function will have either one, two or three distinct solutions, where the function crosses the x-axis. This can be determined by factorising the function: the above example factorises to (x + 4)(2x - 2)(x + 1), giving roots of -4, 1 and -1. The function y = (x - 4)3 has one distinct solution (+4).
The intersection with the y-axis can be obtained by identifying the number part of the function.
1. Sketch the graph of y = (x + 2)(x2 - 9). Mark the points of intersection with the x-axis, and with the y-axis.
The roots of the equation are -2, 3 and -3. The value for the y-axis is obtained by multiplying the integers in the brackets together: 2 x -9 = -18.
2. How many distinct solutions does y = (x - 3)(x2 - x - 6) have?
One solution is given by setting (x - 3) to zero, so 3 is a root. The quadratic can be factorised to (x + 2)(x - 3), giving -2 and 3 as roots. There is already a 3 as a root, so the cubic function has two distinct roots.