To determine the result of vertically shrinking the function [tex]\( f(x) = 2(x+3)^2 \)[/tex] by a factor of [tex]\( \frac{1}{2} \)[/tex] and then reflecting it across the [tex]\( x \)[/tex]-axis, we will proceed with the following steps:
### Step 1: Vertical Shrink by a Factor of [tex]\(\frac{1}{2}\)[/tex]
A vertical shrink of a function [tex]\( f(x) \)[/tex] by a factor of [tex]\( \frac{1}{2} \)[/tex] means that we need to multiply the function by [tex]\( \frac{1}{2} \)[/tex].
The original function is:
[tex]\[ f(x) = 2(x + 3)^2 \][/tex]
After applying the vertical shrink:
[tex]\[ g(x) = \frac{1}{2} \cdot 2(x + 3)^2 \][/tex]
[tex]\[ g(x) = (x + 3)^2 \][/tex]
### Step 2: Reflect Across the [tex]\( x \)[/tex]-Axis
Reflecting a function across the [tex]\( x \)[/tex]-axis involves multiplying the entire function by [tex]\(-1\)[/tex].
The function after the vertical shrink is:
[tex]\[ g(x) = (x + 3)^2 \][/tex]
Reflecting this function across the [tex]\( x \)[/tex]-axis:
[tex]\[ h(x) = - (x + 3)^2 \][/tex]
### Conclusion
The function resulting from first vertically shrinking [tex]\( f(x) = 2(x+3)^2 \)[/tex] by a factor of [tex]\( \frac{1}{2} \)[/tex] and then reflecting it across the [tex]\( x \)[/tex]-axis is:
[tex]\[ h(x) = - (x + 3)^2 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{D \: y = -(x + 3)^2} \][/tex]
