The vertex form of a quadratic function is given by:
�
(
�
)
=
�
(
�
−
ℎ
)
2
+
�
f(x)=a(x−h)
2
+k
where $(h, k)$ is the vertex of the parabola.
Substituting the given vertex $\left(-2,\ -6\right)$, we get:
�
(
�
)
=
�
(
�
−
(
−
2
)
)
2
−
6
f(x)=a(x−(−2))
2
−6
Simplifying:
�
(
�
)
=
�
(
�
+
2
)
2
−
6
f(x)=a(x+2)
2
−6
To find the value of $a$, we use the fact that the function passes through the point $\left(-1,\ 2\right)$:
2
=
�
(
−
1
+
2
)
2
−
6
2=a(−1+2)
2
−6
Solving for $a$:
\begin{align*}
2 + 6 &= a(1)^2 \
a &= 8
\end{align*}
Therefore, the quadratic function in vertex form that satisfies the given conditions is:
�
(
�
)
=
8
(
�
+
2
)
2
−
6
f(x)=8(x+2)
2
−6