Investigation

## Curve $C$ - how can calculus help?

Curve $C$, the red curve, does not look like one of the typical cubic shapes we have seen before. However, it has rotational symmetry about $(0,1)$.

It has some similarities to the curve $y=x^3+1$ but it does not have the same key feature at $x=0$.

What are the features of this curve compared to $y=x^3$ or a translation of it?

We could try to form a set of simultaneous equations to find a cubic equation, but would this be an efficient way to find an equation?

### Calculus of curve $C$

The curve has no stationary points.

If a cubic curve given by a function $y=g(x)$ has no stationary points then what do we know about $g'(x)$?

After using this resource you could take a look at Can you find… curvy cubics edition.