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?