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The graph shown represents a transformation of the basic quadratic function \( f(x) = x^2 \).

### Analyzing the Transformation:

1. **Reflection:** The graph opens downward, indicating a reflection over the x-axis. This means the leading coefficient is negative.

2. **Vertical Stretch/Compression:** The graph appears to be stretched vertically since it is narrower than the graph of \( x^2 \).

3. **Horizontal Translation:** The vertex of the graph is located at \( (2, 4) \), indicating that the graph has been translated 2 units to the right.

4. **Vertical Translation:** The vertex also indicates a translation 4 units up.

### Formulating the Equation:

Given the transformations, the equation can be written as:
\[
g(x) = a(x-h)^2 + k
\]
where \( (h, k) \) is the vertex of the parabola.

Substituting the values \( h = 2 \) and \( k = 4 \), and considering the reflection over the x-axis (which makes \( a \) negative), the equation becomes:
\[
g(x) = a(x-2)^2 + 4
\]

To determine the value of \( a \), observe that when \( x = 0 \), \( g(x) = -5 \):
\[
-5 = a(0-2)^2 + 4
\]
\[
-5 = 4a + 4
\]
\[
4a = -9 \quad \Rightarrow \quad a = -\frac{9}{4}
\]

Thus, the equation of the transformed function is:
\[
g(x) = -\frac{9}{4}(x-2)^2 + 4
\]

Would you like more details on any part of the process? Here are five related questions you might find useful:

1. What are the effects of different values of \(a\) on the shape of a parabola?
2. How does the vertex form of a quadratic function help in identifying transformations?
3. What are other possible transformations of a quadratic function?
4. Can this method be applied to other types of functions, like cubic or exponential?
5. How would the graph change if the quadratic function had a different vertex?

**Tip:** When identifying transformations, always start by locating the vertex and determining if the parabola opens upwards or downwards!

The graph above is a transformation of the function x^2. Write an equation for the function graphed above.

Learn how to write the equation of a quadratic function based on its transformation from the parent function x^2. This problem covers transformations including reflection, translation, and stretching, using the vertex form of the quadratic equation. Suitable for students in Grades 9-12.

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