Question

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

Explanation:

Step1: Start with the given formula

We have the formula \( y = a(x - h)^2 + k \). Our goal is to manipulate this formula to match one of the options. Let's solve for \( a \) first.
Subtract \( k \) from both sides: \( y - k = a(x - h)^2 \)

Step2: Solve for \( a \)

Divide both sides by \( (x - h)^2 \) (assuming \( x
eq h \)): \( a=\frac{y - k}{(x - h)^2} \)
This matches option D. Let's check why other options are incorrect:

• Option A: If we try to solve for \( h \), from \( y - k = a(x - h)^2 \), we get \( (x - h)^2=\frac{y - k}{a} \), then \( x - h=\pm\sqrt{\frac{y - k}{a}} \), so \( h = x\mp\sqrt{\frac{y - k}{a}} \), not the form in A.
• Option B: Solving for \( k \), we get \( k = y - a(x - h)^2 \), not \( y+(x - h)^2 \).
• Option C: Solving for \( x \), from \( y - k = a(x - h)^2 \), we get \( (x - h)^2=\frac{y - k}{a} \), then \( x - h=\pm\sqrt{\frac{y - k}{a}} \), so \( x = h\pm\sqrt{\frac{y - k}{a}} \), not the form in C.

Answer:

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