We have the curve f(x) = (x^2-5x)(x-1)+ 3x. Sketch the graph y=f(x), making sure to plot the co-ordinates where the curve meets the axes.

The first thing we want to do is re-write the curve f(x) in a format in which we can read and easily plot a graph. If we can include the '3x' in the factorised part of the equation, it will be neater and we'll be able to plot it. To do this, we need to expand f(x) to get f(x) = x^3 -6x^2 + 8x and we can see the initial '3x' has been absorbed into the main equation. We can factorise x to obtain f(x) = x(x^2 -6x + 8) to leave us with the quadratic equation which factorises to f(x)=x(x-2)(x-4). We should be able to recognise that the graph will hit the x-axis at 0, 2 and 4, but if not, this comes easily with practise. If we set y=0, we have equations x = 0, x = 2 and x = 4 and we can draw these co-ordinates (0,0), (0,2) and (0,4) on the graph. The curve is a cubic and positive so from our knowledge we can estimate the general shape of the curve.