To solve this problem, we start with the given quadratic equation in vertex form:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
We need to manipulate this equation to find an expression that corresponds to one of the provided options.
### Step-by-Step Solution:
1. Rearranging for [tex]\( a \)[/tex]:
We start by isolating the term that includes [tex]\( a \)[/tex]:
[tex]\[
y = a(x - h)^2 + k
\][/tex]
Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[
y - k = a(x - h)^2
\][/tex]
To solve for [tex]\( a \)[/tex], we divide both sides by [tex]\( (x - h)^2 \)[/tex]:
[tex]\[
a = \frac{y - k}{(x - h)^2}
\][/tex]
This matches the option:
[tex]\[
\text{B. } a = \frac{y - k}{(x - h)^2}
\][/tex]
Let's confirm that the other options do not match.
2. Checking other options:
- Option A:
This option states:
[tex]\[
k = y + (x - h)^2
\][/tex]
However, rearranging our original equation does not lead us to this form.
- Option C:
This option states:
[tex]\[
x = \pm \sqrt{\frac{y - k}{a}} - h
\][/tex]
While this can be derived from our equation, it is quite distinct from a simple rearrangement for [tex]\( a \)[/tex].
- Option D:
This option states:
[tex]\[
h = x - \left( \frac{y - k}{a} \right)^2
\][/tex]
Again, this is a transformation of the original equation, different from solving directly for [tex]\( a \)[/tex].
Upon careful review, the correct transformation of the given quadratic equation to isolate [tex]\( a \)[/tex] matches exactly with option B.
Thus, the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
