Problem

The curve $C_1$ has equation

$$\begin{align*} y=3x^2+6x+9. \end{align*}$$

(a) Write $3x^2+6x+9$ in the form

$$\begin{align*} a(x+b)^2+c \end{align*}$$

where $a$, $b$ and $c$ are constants to be found.

The point $P$ is the minimum point of $C_1$.

(b) Deduce the coordinates of $P$.

A different curve $C_2$ has equation

$$\begin{align*} y=Ax^3+Bx^2+Cx+D \end{align*}$$

where $A$, $B$, $C$ and $D$ are constants.

Given that $C_2$

• passes through $P$
• intersects the $x$-axis at $-4$, $-2$ and $3$

(c) find, making your method clear, the values of $A$, $B$, $C$ and $D$.