Question
The diagram shows a graph with equation $y=ax^3+bx^2+cx+d$, where $a$, $b$, $c$, and $d$ are real constants.
The graph passes through the points $(-6,0)$ and $(-2,32)$ and touches the $x$-axis at the point $(6,0)$.
A student attempts to find the equation of the curve, and writes the following working:
$\boxed {y=(x+6)(x-6)^2 \\ y=(x+6)(x^2-12x+36) \\ y=x^3-6x^2-36x+216}$
a) Explain the mistake the student has made.
b) Find the correct equation of the curve.
Best Answer
The only thing missing is something that you have mentioned: the coefficient $a$. It is clear that $y=a(x+6)(x-6)^2$ and then $x=-2\implies y=256a$. So, $a$ should be equal to $\frac18$.