Review question

# Can we show these two cubic curves touch? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8543

## Interactive graph

A curve is given by the equation $\begin{equation*} y = ax^3 - 6ax^2 + (12a + 12)x - (8a + 16), \label{eqi:1}\tag{*} \end{equation*}$ where $a$ is a real number. Show that this curve touches the curve with equation $\begin{equation*} y = x^3 \label{eqi:2}\tag{*{*}} \end{equation*}$

at $(2,8)$.

You might like to use this applet to explore the behaviour of the curve $\eqref{eqi:1}$ as $a$ varies. The blue curve is $y=x^3$ $\eqref{eqi:2}$.

You can also zoom in or out if that helps to see the behaviour of the curves.

As $a$ varies, you might notice the following.

• The curves always touch at $(2,8)$ as the question suggests.
• What happens to the red curve when $a=0$?
• Are there any values of $a$ for which the curves do not intersect again?