To determine which equations are equivalent to the given formula [tex]\( y = a(x - h)^2 + k \)[/tex], let's analyze each option step-by-step:
### Option A: [tex]\( h = x - \sqrt{\left(\frac{y - k}{a}\right)} \)[/tex]
Starting with the formula:
[tex]\[ y = a(x - h)^2 + k \][/tex]
We need to solve this for [tex]\( h \)[/tex].
1. Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[ y - k = a(x - h)^2 \][/tex]
2. Divide both sides by [tex]\( a \)[/tex]:
[tex]\[ \frac{y - k}{a} = (x - h)^2 \][/tex]
3. Take the square root of both sides:
[tex]\[ \sqrt{\frac{y - k}{a}} = x - h \][/tex]
4. Solve for [tex]\( h \)[/tex]:
[tex]\[ h = x - \sqrt{\frac{y - k}{a}} \][/tex]
Thus, Option A is correct.
### Option B: [tex]\( a = \frac{y - k}{(x - h)^2} \)[/tex]
Starting with the formula:
[tex]\[ y = a(x - h)^2 + k \][/tex]
We need to solve this for [tex]\( a \)[/tex].
1. Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[ y - k = a(x - h)^2 \][/tex]
2. Divide both sides by [tex]\( (x - h)^2 \)[/tex]:
[tex]\[ a = \frac{y - k}{(x - h)^2} \][/tex]
Thus, Option B is correct.
### Option C: [tex]\( k = y +(x - h)^2 \)[/tex]
Starting with the formula:
[tex]\[ y = a(x - h)^2 + k \][/tex]
We need to solve this for [tex]\( k \)[/tex].
1. 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 states [tex]\( k = y + (x - h)^2 \)[/tex], which is incorrect according to our solution.
Thus, Option C is incorrect.
### Option D: [tex]\( x = \pm \sqrt{\frac{y - k}{a}} + h \)[/tex]
Starting with the formula:
[tex]\[ y = a(x - h)^2 + k \][/tex]
We need to solve this for [tex]\( x \)[/tex].
1. Subtract [tex]\( k \)[/tex] from both sides:
[tex]\[ y - k = a(x - h)^2 \][/tex]
2. Divide both sides by [tex]\( a \)[/tex]:
[tex]\[ \frac{y - k}{a} = (x - h)^2 \][/tex]
3. Take the square root of both sides:
[tex]\[ \sqrt{\frac{y - k}{a}} = x - h \][/tex]
or
[tex]\[ -\sqrt{\frac{y - k}{a}} = x - h \][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \sqrt{\frac{y - k}{a}} + h \][/tex]
or
[tex]\[ x = -\sqrt{\frac{y - k}{a}} + h \][/tex]
Thus, Option D is correct.
### Conclusion
The correct options are:
- Option A: [tex]\( h = x - \sqrt{\left(\frac{y - k}{a}\right)} \)[/tex]
- Option B: [tex]\( a = \frac{y - k}{(x - h)^2} \)[/tex]
- Option D: [tex]\( x = \pm \sqrt{\left(\frac{y - k}{a}\right)} + h \)[/tex]
So, the correct answer is:
[tex]\[ \text{Options: } [1, 2, 4] \][/tex]
