1. General differences between quadratics and cubics

I am trying to summarise the differences between a general quadratic and general cubic. So far i have this:

- A quadratic is not guaranteed to have real roots but a cubic must always have at least one
- A quadratic will always have a turning point but a cubic is not guaranteed to
- A quadratic will never have a point of inflexion but a cubic always will?

Do others agree with this ( particularly the last one) and what other differences are there?
I think there might be something about the shape but i don't know how to summarise it....

Thoughts welcome...

2. Re: General differences between quadratics and cubics

another thought... something to do with the range of a quadratic compared to a cubic?

3. Re: General differences between quadratics and cubics

The range of a cubic is always "all real numbers". The range of a quadratic never is.

4. Re: General differences between quadratics and cubics

There is such a thing as a standard quadratic

$y = x^2$

any parabola can be obtained from this standard parabola by scaling and translating (ignore rotations for the moment)

$y=a(x-h)^2 + k$

This isn't true for cubics. Sure there is

$y=x^3$

but there will be cubic curves that are not of the form

$y = a(x-h)^3 + k$

So a cubic curve has a richness of variety that doesn't exist for a quadratic one.

5. Re: General differences between quadratics and cubics

Originally Posted by romsek
There is such a thing as a standard quadratic

$y = x^2$

any parabola can be obtained from this standard parabola by scaling and translating (ignore rotations for the moment)

$y=a(x-h)^2 + k$

This isn't true for cubics. Sure there is

$y=x^3$

but there will be cubic curves that are not of the form

$y = a(x-h)^3 + k$

So a cubic curve has a richness of variety that doesn't exist for a quadratic one.
An interesting observation which i have never thought about! Thanks.