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?
