Question

select the correct answer.
which equation is equivalent to the formula below?
y = a(x - h)² + k

a. h = x - (\frac{y - k}{a})²

b. a = \frac{y - k}{(x - h)²}

c. x = ±√\frac{y - k}{a} - h

d. k = y + (x - h)²

Explanation:

Step1: Start with the given formula

We have the formula \( y = a(x - h)^2 + k \). We need to manipulate this formula to check which of the options is equivalent. Let's try to solve for \( a \) first.

Step2: Isolate the term with \( a \)

Subtract \( k \) from both sides of the equation:
\( y - k = a(x - h)^2 \)

Step3: Solve for \( a \)

Divide both sides of the equation by \( (x - h)^2 \) (assuming \( x
eq h \) so that we don't divide by zero):
\( a=\frac{y - k}{(x - h)^2} \)
This matches option B. Let's just verify the other options to be sure.

For option A: Starting from \( y=a(x - h)^2 + k \), if we try to solve for \( h \), we would have \( y - k=a(x - h)^2\), then \( \frac{y - k}{a}=(x - h)^2 \), then \( x - h=\pm\sqrt{\frac{y - k}{a}} \), then \( h=x\mp\sqrt{\frac{y - k}{a}} \), which is not the same as option A.

For option C: Solving \( y=a(x - h)^2 + k \) for \( x \), we get \( y - k=a(x - h)^2 \), \( \frac{y - k}{a}=(x - h)^2 \), \( x - h=\pm\sqrt{\frac{y - k}{a}} \), \( x=h\pm\sqrt{\frac{y - k}{a}} \), which is not the same as option C.

For option D: Solving \( y=a(x - h)^2 + k \) for \( k \), we get \( k=y - a(x - h)^2 \), which is not the same as option D.

Answer:

B. \( a = \frac{y - k}{(x - h)^2} \)