Lesson BASICS - sketching Cubics

Source code of 'BASICS - sketching Cubics'

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About longjonsilver: I have a new job in September, teaching

Introduction

A cubic equation is of the form {{{y=ax^3+bx^2+cx+d}}}. Its general shape is either:

{{{graph(200,200,-2,2,-7,7,x^3-x)}}} or {{{graph(200,200,-2,2,-7,7,-(x^3-x))}}}

Again, like a quadratic, we know the general shape just by looking at the equation.

If we have a +{{{x^3}}} term then the shape orientation is like the first graph above.
If we have a -{{{x^3}}} term then the shape orientation is like the second graph above.

3 ROOTS
As for the turning points, a cubic in general has 3 roots (ie it crosses the x-axis at 3 points). For it to do this, it has to change direction twice, ie 2 turning points... 2 "humps", as seen in the general cubic graphs above.

As the cubic equation changes in content, so the turning points either move and/or get flattened out.

2 ROOTS
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.

EXAMPLE the cubic {{{y=4x^3+8x^2+4x}}}, when factorised gives y=4(x+1)(x+1)(x). In other words, it has 2 roots, at x=0 and x=-1, but still the "2 humped" shape, as {{{graph(200,200,-2,1,-3,5,4(x^3+2x^2+x))}}}

1 ROOT
Finally, reducing the number of roots further, to one, for the cubic equation, the humps disappear altogether and we end up with something like {{{y=5x^3+2x^2+x}}}, which looks like {{{graph(200,200,-1,1,-3,5,5x^3+2x^2+x)}}}

And the simplest cubic is {{{y=x^3}}}, which looks like {{{graph(200,200,-2,2,-9,9,x^3)}}}

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An "Oddity"
EXAMPLE What of the cubic {{{y=4x^3+8x^2+4x-1}}}? This looks like {{{graph(200,200,-2,1,-4,3,4(x^3+2x^2+x)-1)}}}

This means we still have 2 turning points but only one root.

When you are given an equation like this, you really must have more ammo in your mathematical arsenal, such as differentiation, which will help you find the turning points, while factorisation will give you the one root. Your sketch will then use all this info. so you can sketch it correctly.

However, for sketching "basic" cubics, you should be given "nice" equations.
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In Summary
A cubic equation must have 1, 2 or 3 solutions/roots. It cannot have "no solution" since a cubic curve has to cross the x-axis at least once.

Depending upon the number of roots, this tells you something of the shape:
 --> 3 roots means the curve crosses the x-axis 3 times, which forces the curve to have 2 turning points.
 --> 2 roots means the curve crosses/touches the x-axis 2 times.

In all of this, you have to practice lots to see all the possibilities.