Question

Given $y=a\left( x-2 \right)\left( x+4 \right)$. In the given quadratic equation, $a$ is a nonzero constant. the graph of the equation in the xy-plane is a parabola with vertex $\left( c,d \right)$. Which of the following is equal to $d$?
(a) $-9a$
(b) $-8a$
(c) $-5a$
(d) $-2a$

Answer 1

Hint: We will modify the equation and then use a change of variables to obtain an equation in the standard form for a parabola. The standard parabola with the standard equation has the vertex coordinates $\left( 0,0 \right)$. We will use this fact to calculate the coordinates of the parabola represented by the given equation.

Complete step by step answer:
The given equation is $y=a\left( x-2 \right)\left( x+4 \right)$. We will modify this quadratic equation. We have
$y=a\left( {{x}^{2}}+2x-8 \right)$.
Now, in the above equation, we will write $-8$ as $+1-9$. So, we get the following,
$\begin{align}
  & y=a\left( {{x}^{2}}+2x+1-9 \right) \\
 & =a\left( {{\left( x+1 \right)}^{2}}-9 \right)
\end{align}$
Next, we will rearrange the above equation so that we have the square term on one side and the remaining terms on the other side as follows,
$\begin{align}
  & \dfrac{y}{a}={{\left( x+1 \right)}^{2}}-9 \\
 & \therefore {{\left( x+1 \right)}^{2}}=\dfrac{y}{a}+9 \\
\end{align}$
Now, we will change the variables in the following manner: let $X=x+1$ and let $Y=\dfrac{y}{a}+9$.
So, the given equation has now become ${{X}^{2}}=Y$. This equation represents the standard parabola with the vertex at origin. Since the vertex is at the origin, we have $\left( X,Y \right)=\left( 0,0 \right)$.
Substituting the values of the changed variables, we get $x+1=0$ and $\dfrac{y}{a}+9=0$. Solving these two equations for $x$ and $y$, we get $x=-1$ and $y=-9a$.
Hence, the coordinates of the vertex of the given equation representing parabola are $\left( -1,-9a \right)$.
Therefore, the correct option is (a).

Note:
 It is necessary that we know the standard equations representing conic sections. Knowing these equations makes it easier to modify the given equation into the standard form. For the standard equations, we know how to find the coordinates of foci or vertex and equations for directrix, etc.