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.