We are given three functions: $f(x) = -\frac{1}{2}x^2$, $z(x) = x - 4$, and $p(x) = x + 5$. First, we need to find the composite function $z(f(p(x)))$. Then, we need to describe the transformations applied to the parent function $x^2$.

1. Problem Description

We are given three functions: $f(x) = -\frac{1}{2}x^2$, $z(x) = x - 4$, and $p(x) = x + 5$.

First, we need to find the composite function $z(f(p(x)))$.

Then, we need to describe the transformations applied to the parent function $x^2$.

2. Solution Steps

First, we find $p(x)$:

$p(x) = x + 5$

Next, we find $f(p(x))$:

$f(p(x)) = f(x+5) = -\frac{1}{2}(x+5)^2$

Finally, we find $z(f(p(x)))$:

$z(f(p(x))) = z(-\frac{1}{2}(x+5)^2) = -\frac{1}{2}(x+5)^2 - 4$

$z(f(p(x))) = -\frac{1}{2}(x^2 + 10x + 25) - 4 = -\frac{1}{2}x^2 - 5x - \frac{25}{2} - 4 = -\frac{1}{2}x^2 - 5x - \frac{25}{2} - \frac{8}{2} = -\frac{1}{2}x^2 - 5x - \frac{33}{2}$

Now, let's analyze the transformations to the parent function $x^2$.

The expression is $z(f(p(x))) = -\frac{1}{2}(x+5)^2 - 4$.

Reflection: Since the coefficient of the $x^2$ term is negative, there is a reflection over the x-axis.

Stretch/Compression: The coefficient is $-\frac{1}{2}$, so there is a vertical compression by a factor of $\frac{1}{2}$.

Vertical Shift: There is a vertical shift down by 4 units because of the "-4" term.

Horizontal Shift: There is a horizontal shift to the left by 5 units because of the "(x+5)" term.

3. Final Answer

1) $z(f(p(x))) = -\frac{1}{2}(x+5)^2 - 4$

Reflection: Reflection over the x-axis

Stretch/Compression: Vertical compression

Vertical Shift: Vertical shift down

Horizontal Shift: Horizontal shift left