https://file.aiquickdraw.com/mathbot/file/1727640984667nbf9gh1d.jpg

We are given two functions, \( f(x) = x^2 \) (shown in black) and \( g(x) \) (shown in blue), and it is stated that \( g(x) \) is derived from \( f(x) \) through a sequence of transformations. 

Let's analyze the transformations from the graph:

1. **Horizontal translation**: The function \( g(x) \) appears to have been shifted to the right. The vertex of \( f(x) = x^2 \) is at \( (0, 0) \), while the vertex of \( g(x) \) is at \( (3, 5) \). This indicates a horizontal shift by \( 3 \) units to the right.

2. **Vertical translation**: In addition to the horizontal shift, the vertex of \( g(x) \) is also shifted up by \( 5 \) units. 

3. **Reflection**: The graph of \( g(x) \) appears upside down compared to \( f(x) \), indicating a reflection across the x-axis.

4. **Vertical compression**: The graph of \( g(x) \) is also "narrower" compared to the parabola \( f(x) = x^2 \), which indicates a vertical compression.

### Summary of transformations:
- Horizontal shift by \( 3 \) units to the right: \( f(x - 3) \)
- Vertical shift by \( 5 \) units up: \( f(x - 3) + 5 \)
- Reflection across the x-axis: Multiply by \( -1 \): \( -f(x - 3) + 5 \)
- Vertical compression (assuming by a factor of \( \frac{1}{2} \)): \( \frac{1}{2}(-f(x - 3)) + 5 \)

Thus, the equation for \( g(x) \) is:

\[
g(x) = -\frac{1}{2}(x - 3)^2 + 5
\]

This equation reflects the sequence of transformations applied to \( f(x) = x^2 \).

Would you like any further details or clarifications?

Here are 5 related questions you might find useful:
1. How do vertical and horizontal shifts affect the graph of a quadratic function?
2. What is the general form of a quadratic function that has been reflected, shifted, and stretched?
3. How does the vertex form of a parabola help identify transformations?
4. What role does the coefficient in front of the squared term play in a quadratic equation?
5. How do we determine if a function is compressed or stretched vertically?

**Tip**: When analyzing transformations, always start by identifying changes in the vertex, as these often help pinpoint shifts and reflections in the function.

Learn how to find the equation for g(x) by applying a series of transformations to f(x) = x^2, including horizontal and vertical shifts, reflection across the x-axis, and vertical compression. This step-by-step guide is suitable for high school students in Grades 9-11.

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