2021-I-10

$\begin{array}{rcl}
f(-3) & = & 0 \\
k_1 + k_2(-3+4)^2 & = & 0 \\
k_1 + k_2 & = & 0 \ \ldots\unicode{x2460}
\end{array}$

$\begin{array}{rcl}
f(2) & = & 105 \\
k_1 + k_2(2+4)^2 & = & 105 \\
k_1 + 36k_2 & = & 105 \ \ldots\unicode{x2461}
\end{array}$

$\unicode{x2461} – \unicode{x2460}$, we have

$\begin{array}{rcl}
35k_2 & = & 105 \\
k_2 & = & 3
\end{array}$

Sub $k_2=3$ into $\unicode{x2460}$, we have

$\begin{array}{rcl}
k_1 + 3 & = & 0 \\
k_1 & = & -3
\end{array}$

$\therefore f(x) = -3+3(x+4)^2$.

$\begin{array}{rcl}
f(0) & = & -3 + 3(0+4)^2 \\
& = & 45
\end{array}$

1. For the $y$-intercept of $G$, sub. $x=0$ into the equation of $G$, we have

   $\begin{array}{rcl}
   y & = & f(0) + 3 \\
   & = & 45 + 3 \\
   & = & 48
   \end{array}$

   $\therefore$ the $y$-intercept of $G$ is $48$.

2. For the $x$-intercept of $G$, sub. $y=0$ into the equation of $G$, we have

   $\begin{array}{rcl}
   f(x) + 3 & = & 0 \\
   -3 + 3(x+4)^2 + 3 & = & 0 \\
   (x+4)^2& = & 0 \\
   \end{array}$

   $\therefore x=4$ (repeated).

   $\therefore$ the $x$-intercept of $G$ is $4$.