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.

Question

The dish is rotated by reflecting it over the x-axis.
Write the equation of the transformed function.

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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) .

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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:

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