https://file.aiquickdraw.com/mathbot/file/1727064499019cct69zzx.jpg

The graph shown in the image is a downward-opening parabola, which is a transformation of the function \( f(x) = x^2 \).

We can determine the transformed function \( g(x) \) by analyzing the graph:

1. **Reflection**: The graph opens downward, which means there is a vertical reflection across the x-axis. This is represented by multiplying the function by \( -1 \), giving \( -x^2 \).

2. **Vertical Shift**: The vertex of the graph appears to be at \( (0, -4) \), which indicates the function has been shifted down by 4 units. This adds \( -4 \) to the function.

Thus, the equation for the function \( g(x) \) is:
\[
g(x) = -x^2 - 4
\]

Let me know if you need more details or explanations.

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Here are 5 related questions you can explore:
1. What is the effect of horizontal shifts on the function \( f(x) = x^2 \)?
2. How would the equation change if the graph had a vertical stretch or compression?
3. How do you determine the domain and range of a parabola like the one in the graph?
4. What would the function be if the vertex were shifted to \( (2, -4) \)?
5. How would you describe the transformation if the parabola opened upward instead?

**Tip**: Always check the vertex and orientation of a parabola to identify vertical shifts and reflections when analyzing transformations.

The graph above is a transformation of the function x^2. Give the function in the graph above.

Learn how the quadratic function f(x) = x^2 transforms into g(x) = -x^2 - 4 through reflection and vertical shift. This problem covers key algebraic transformations for Grades 9-11.

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