This means we still have 2 turning points but only one root. In other words, it has 2 roots, at x=0 and x=-1, but still the "2 humped" shape, asįinally, reducing the number of roots further, to one, for the cubic equation, the humps disappear altogether and we end up with something like, which looks likeĪnd the simplest cubic is, which looks likeĮXAMPLE What of the cubic ? This looks like If we move the cubic curve up/down, then at certain positions, one of the turning points just touches the x-axis, and we then have just 2 roots: in truth, 2 roots have "become the same" ie looking at the algebra, we will have duplicated roots.ĮXAMPLE the cubic, when factorised gives y=4(x+1)(x+1)(x). 2 "humps", as seen in the general cubic graphs above.Īs the cubic equation changes in content, so the turning points either move and/or get flattened out.
For it to do this, it has to change direction twice, ie 2 turning points. If we have a - term then the shape orientation is like the second graph above.Īs for the turning points, a cubic in general has 3 roots (ie it crosses the x-axis at 3 points). If we have a + term then the shape orientation is like the first graph above. Its general shape is either:Īgain, like a quadratic, we know the general shape just by looking at the equation.
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