GCSE(F), GCSE(H),

A cubic function will have an *x*^{3} term. An example of a cubic function is 2*x*^{3} + 8*x*^{2} - 2*x* - 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)(2*x* - 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)(*x*^{2} - 9). Mark the points of intersection with the *x*-axis, and with the *y*-axis.

Answer:

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)(*x*^{2} - *x* - 6) have?

Answer: 2

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.

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