Answer
\[
3 x^{2}-2 x+7=Y
\]
Work Step by Step
The derivative is the slope of the tangent line to $y$ at $x$
\[
a x^{2}+b x+c=y
\]
\[
\begin{array}{l}
2 a x+b=y^{\prime} \\
2 a(1)+b=4\quad \text { slope } 4 \text { at } 1=x
\end{array}
\]
$2 a(-1)+b=-8 \quad$ slope -8 at $-1=x$
Solve the system
\[
\begin{aligned}
&2 a+b=4 \\
&=-2 a+b=-8
\end{aligned}
\]
Add [2] to [1]
\[
2 a-2 a+b+b=4-8
\]
\[
\begin{array}{r}
2 b=-4 \\
-2=b
\end{array}
\]
Use either equation to find $a$
\[
(-2)+2 a=4
\]
\[
\begin{array}{l}
2 a=6 \\
3=a
\end{array}
\]
The parabola passes through (2,15)
\[
\begin{array}{l}
a(2)^{2}+b(2)+c=15 \\
4 a+2 b+c=15
\end{array}
\]
Plug in the $a$ , $b$ we found
\[
\begin{aligned}
4(3)+2(-2)+c =15\\
8+c=15 \\
7=c
\end{aligned}
\]
The parabola is
\[
3 x^{2}-2 x+7=y
\]
