Cubic Function

L2MAEL

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The cubic function. Yes, that’s the ‘S’-shaped graph.

A cubic function has an exponent of 3, as you can tell by the name:

$y=x^3$

The equation of a cubic is this:

$y=a{x}^3+b{x}^2+cx+d$

However, we can’t translate that effectively, so we have to use:

Respectively the inflection form, and the intercept form.

$y=a(x-p{)}^3+q$

$y=a(x-\alpha )(x-\beta )(x-\gamma )$

..where $(p,q)$ is the point of inflection. Point of inflections are a point just like the vertex of a quadratic function. It’s where the concavity (similar concepts to gradient) changes.

it’s very similar to the quadratic function — just the exponent swapped.

..The $\alpha$, $\beta$, $\gamma$ stands for the intercepts along the $x$-axis.

Let’s look at this graph:

$y=2(x-1{)}^3+2$

Function is drawn using the inflection form.

As we can see, the cubic function has a point of inflection at $(1,2)$ marked in pink — since the inflection point’s equation states that $(p,q)$ is the point of inflection.

Moving onto the next form, we have the intercept form. Unline the inflection point form, the intercept form has a local maximum & minimum (local extrema). This is because when you expand the intercept form, you get a $x^2$. The $x^2$ makes it so that the cubic have a local max & min (more explanation on calculus using derivatives).

$y=(x-1)(x-2)(x-3)$

Function is drawn using the intercept form.

As you can see, the graph intercepts the $x$-axis at exactly $(1,0)$ $(2,0)$, and $(3,0)$.

Local maximum and minimums are the points in orange in the image below:

If you draw a gradient (just a straight) along the cubic function:

You can see that the gradient becomes 0, as soon as it meets the local maximum and minimum. As you can tell by the name, a local maximum and minimum are the max & min points of a cubic function.

Trends:

Trends are the ones you saw in stats (if you have done it). Is it an increasing trend? Or, a decreasing trend?

We read trends from left to right — like you read a book. On the graphs above, both graphs have an increasing trend, because $a$, the scale factor is in the positives.

If a function has a negative scale factor $(a)$, then you get a negative trend (try it on Desmos).