We are given three functions: $f(x) = \frac{1}{5}x^2$, $p(x) = -x$, and $z(x) = x + 8$. We need to find the composition $z(f(p(x)))$. Then, we need to describe the transformations from the parent function $x^2$.

1. Problem Description

We are given three functions: $f(x) = \frac{1}{5}x^2$, $p(x) = -x$, and $z(x) = x + 8$. We need to find the composition $z(f(p(x)))$. Then, we need to describe the transformations from the parent function $x^2$.

2. Solution Steps

First, we find $p(x) = -x$.

Next, we find $f(p(x))$ by substituting $p(x)$ into $f(x)$:

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

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

$z(f(p(x))) = z(\frac{1}{5}x^2) = \frac{1}{5}x^2 + 8$.

Now let's discuss the transformations.

The parent function is $x^2$.

The transformed function is $\frac{1}{5}x^2 + 8$.

Compared to the parent function $x^2$:

* Reflection: No reflection across the x-axis since the coefficient of $x^2$ is positive.

* Stretch/Compression: Vertical compression by a factor of $\frac{1}{5}$.

* Vertical Shift: Upward shift by 8 units.

* Horizontal Shift: No horizontal shift.

3. Final Answer

$z(f(p(x))) = \frac{1}{5}x^2 + 8$

Reflection: None

Stretch/Compression: Vertical Compression

Vertical Shift: Up

Horizontal Shift: None