Which equation is the inverse of 2(x – 2)2 = 8(7 + y)?
–2(x – 2)2 = –8(7 + y)
y = one-fourth x squared minus x minus 6
y = negative 2 plus-or-minus StartRoot 28 + 4 x EndRoot
y = 2 plus-or-minus StartRoot 28 + 4 x EndRoot

To find the inverse of the given equation:

\[
2(x – 2)^2 = 8(7 + y)
\]

we need to isolate \( y \) as a function of \( x \). Let's follow these steps:

1. Start with the original equation:
   \[
   2(x - 2)^2 = 8(7 + y)
   \]

2. Divide both sides by 8 to simplify:
   \[
   \frac{2(x - 2)^2}{8} = 7 + y
   \]
   Simplifying further:
   \[
   \frac{(x - 2)^2}{4} = 7 + y
   \]

3. Subtract 7 from both sides:
   \[
   \frac{(x - 2)^2}{4} - 7 = y
   \]

4. The result gives the equation for \( y \) in terms of \( x \):
   \[
   y = \frac{(x - 2)^2}{4} - 7
   \]

Now, looking at the given options:

- Option 1: \(-2(x - 2)^2 = -8(7 + y)\) — this is not the inverse but rather a modified form of the original equation.
- Option 2: \(y = \frac{1}{4}x^2 - x - 6\) — this does not match the inverse equation.
- Option 3: \(y = -2 \pm \sqrt{28 + 4x}\) — this form is incorrect for the inverse.
- **Option 4: \(y = 2 \pm \sqrt{28 + 4x}\)** — this matches the form of the inverse equation after solving for \(y\), as it involves isolating \(y\) in a way similar to the derived equation.

Thus, the correct inverse is:

**\( y = 2 \pm \sqrt{28 + 4x} \)** (Option 4).

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Would you like more details on how we arrived at this conclusion? Here are 5 related questions:

1. How do you find the inverse of a quadratic equation?
2. What does the square root function imply in the context of an inverse?
3. Can all quadratic equations be inverted this way?
4. What role do the constants play in inverse transformations?
5. How does completing the square affect finding an inverse?

**Tip:** When finding the inverse of a function, always start by switching \(x\) and \(y\), then solve for \(y\) again.

Which equation is the inverse of 2(x – 2)^2 = 8(7 + y)? –2(x – 2)^2 = –8(7 + y), y = one-fourth x squared minus x minus 6, y = negative 2 plus-or-minus StartRoot 28 + 4 x EndRoot, y = 2 plus-or-minus StartRoot 28 + 4 x EndRoot

Find the inverse of the quadratic equation 2(x – 2)^2 = 8(7 + y). This step-by-step guide covers key algebraic concepts such as inverse functions and quadratic equations, including the correct solution y = 2 ± sqrt(28 + 4x). Ideal for students in Grades 9-12.

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