Question

A satellite dish is modeled by the parabolic function in Standard Form h(x)=-0.25x2+4, or in Vertex Form h(x)=-0.25(x-0)2+4, where x is the horizontal distance from the center of the dish.
The dish is rotated by reflecting it over the x-axis.
Write the equation of the transformed function.

_________________________



The dish is moved up by 5 units to increase its height. Write the equation of the new function combined with the equation in A) .

__________________________

Answer 1

To solve the problem, we'll first address the reflection over the x-axis and then the upward translation by 5 units.

### A) Reflecting the function over the x-axis

The original function is given in standard form as:

\[
h(x) = -0.25x^2 + 4
\]

When a function is reflected over the x-axis, the sign of the function is changed. Therefore, the equation of the transformed function after reflection is:

\[
h_{\text{reflected}}(x) = -(-0.25x^2 + 4) = 0.25x^2 - 4
\]

### B) Moving the dish up by 5 units

To move the dish up by 5 units, we simply add 5 to the reflected function:

\[
h_{\text{final}}(x) = 0.25x^2 - 4 + 5
\]

So the new equation becomes:

\[
h_{\text{final}}(x) = 0.25x^2 + 1
\]

### Final Answers

1. The equation of the transformed function after reflecting it over the x-axis is:

\[
h_{\text{reflected}}(x) = 0.25x^2 - 4
\]

2. The combined equation after moving the dish up by 5 units is:

\[
h_{\text{final}}(x) = 0.25x^2 + 1
\]