To determine which equations are equivalent to the given formula: [tex]\( y = a(x-h)^2 + k \)[/tex], let's analyze each option one by one.
### Option A: [tex]\( h = x - \sqrt{\left(\frac{y - k}{a}\right)} \)[/tex]
Given the original formula [tex]\( y = a(x-h)^2 + k \)[/tex], to solve for [tex]\( h \)[/tex]:
[tex]\[ y = a(x-h)^2 + k \][/tex]
Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[ y - k = a(x-h)^2 \][/tex]
Divide both sides by [tex]\( a \)[/tex]:
[tex]\[ \frac{y - k}{a} = (x-h)^2 \][/tex]
Take the square root of both sides:
[tex]\[ \sqrt{\frac{y - k}{a}} = |x - h| \][/tex]
So, [tex]\( x - h \)[/tex] could be either [tex]\( \sqrt{\frac{y - k}{a}} \)[/tex] or [tex]\( -\sqrt{\frac{y - k}{a}} \)[/tex]. Hence, the equation for [tex]\( h \)[/tex] cannot be isolated in the form presented in Option A. Thus, Option A is not correctly solved for [tex]\( h \)[/tex].
### Option B: [tex]\( x = \pm \sqrt{\frac{y - k}{a}} + h \)[/tex]
To solve for [tex]\( x \)[/tex] from the original formula [tex]\( y = a(x-h)^2 + k \)[/tex]:
[tex]\[ y = a(x-h)^2 + k \][/tex]
Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[ y - k = a(x-h)^2 \][/tex]
Divide both sides by [tex]\( a \)[/tex]:
[tex]\[ \frac{y - k}{a} = (x-h)^2 \][/tex]
Take the square root of both sides:
[tex]\[ \sqrt{\frac{y - k}{a}} = |x - h| \][/tex]
Thus, [tex]\( x - h = \pm \sqrt{\frac{y - k}{a}} \)[/tex]
[tex]\[ x = \pm \sqrt{\frac{y - k}{a}} + h \][/tex]
Therefore, Option B is correctly solved for [tex]\( x \)[/tex].
### Option C: [tex]\( k = y + (x-h)^2 \)[/tex]
Given the original formula [tex]\( y = a(x-h)^2 + k \)[/tex], to solve for [tex]\( k \)[/tex]:
[tex]\[ y = a(x-h)^2 + k \][/tex]
Subtract [tex]\( a(x-h)^2 \)[/tex] from both sides:
[tex]\[ y - a(x-h)^2 = k \][/tex]
So,
[tex]\[ k = y - a(x-h)^2 \][/tex]
Option C is not correctly solved for [tex]\( k \)[/tex] because it does not match this derived equation.
### Option D: [tex]\( a = \frac{y - k}{(x-h)^2} \)[/tex]
To solve for [tex]\( a \)[/tex] from the original formula [tex]\( y = a(x-h)^2 + k \)[/tex]:
[tex]\[ y = a(x-h)^2 + k \][/tex]
Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[ y - k = a(x-h)^2 \][/tex]
Divide both sides by [tex]\((x-h)^2\)[/tex]:
[tex]\[ a = \frac{y - k}{(x-h)^2} \][/tex]
Thus, Option D is correctly solved for [tex]\( a \)[/tex].
### Conclusion
The correct solutions are:
- Option B: [tex]\( x = \pm \sqrt{\frac{y - k}{a}} + h \)[/tex]
- Option D: [tex]\( a = \frac{y - k}{(x-h)^2} \)[/tex]
Therefore, the correct answers are options 2 (B) and 4 (D).
