To determine the inverse of the given equation [tex]\( 2(x-2)^2 = 8(7+y) \)[/tex], you can follow these steps:
1. Start with the given equation:
[tex]\[
2(x-2)^2 = 8(7 + y)
\][/tex]
2. Isolate the term containing [tex]\( y \)[/tex]:
[tex]\[
\frac{2(x-2)^2}{8} = 7 + y
\][/tex]
3. Simplify the left-hand side:
[tex]\[
\frac{(x-2)^2}{4} = 7 + y
\][/tex]
4. Isolate [tex]\( y \)[/tex]:
[tex]\[
y = \frac{(x-2)^2}{4} - 7
\][/tex]
Thus, the inverse equation of [tex]\( 2(x-2)^2 = 8(7+y) \)[/tex] is:
[tex]\[
y = \frac{(x-2)^2}{4} - 7
\][/tex]
Looking at the options available:
1. [tex]\( -2(x-2)^2 = -8(7 + y) \)[/tex]
2. [tex]\( y = \frac{1}{4} x^2 - x - 6 \)[/tex]
3. [tex]\( y = -2 \pm \sqrt{28 + 4x} \)[/tex]
4. [tex]\( y = 2 \pm \sqrt{28 + 4x} \)[/tex]
The correct equation for the inverse is:
[tex]\[
y = \frac{(x-2)^2}{4} - 7
\][/tex]
This matches none of the options verbatim, but it's closest in form to the following (which seems to be derived from a simple term rearrangement/expansion):
[tex]\[
y = -6 + \left(\frac{x}{4}\right)^2 - \frac{x}{2} + 1
\][/tex]
Hence, identifying the correct matching option would require a correct symbolic rearrangement, but the core transformation for the inverse leading up to \\
[tex]\[y = \frac{(x-2)^2}{4} - 7\][/tex].
