The graph crosses the x-axis at the roots of the equation, and this will help us to identify the factors:

$x = - 2, x = - 1 . x = 4$

This means that the brackets must be

$f (x) = (x + 2) (x + 1) (x - 4)$

correct brackets [1]

If we expand the brackets, we get

$f (x) = x^{3} - x^{2} - 10 x - 8$

We can see from the graph that it has a y-intercept of 24, therefore, we need to multiply the whole function by 3 to give the positive cubic graph.

for multiplying by 3 [1]

Wlz3b-Cc_oct-2020-paper-11-graph-q1

The graph of $y = |p (x)|$ means means anything below the x-axis gets reflected above the x-axis.

We could also multiply the function by -3 to give the negative cubic graph.

oct-2020-paper-11-graph-q1-2

As before, the graph of $y = |p (x)|$ means anything below the x-axis gets reflected above the x-axis.

Therefore, the final expression for the function would be

$p (x) = \pm 3 (x + 2) (x + 1) (x - 4)$ [1]

Equations, Inequalities & Graphs (Cambridge O Level Additional Maths)

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