The image shows the graph of a transformation of the function $f(x) = x^2$, with a new function $g(x)$. Based on the graph, it seems that the function $g(x)$ represents a horizontal shift of the original function $f(x) = x^2$.

Here’s the breakdown:

1. The vertex of $f(x) = x^2$ is originally at $(0, 0)$.
2. In the graph provided, the vertex of $g(x)$ appears to be at $(3, 4)$.

This suggests a horizontal shift 3 units to the right and a vertical shift 4 units upwards. Therefore, the transformation applied to $f(x)$ can be written as:

$g(x) = (x - 3)^2 + 4$

This equation represents the function that has been shifted horizontally and vertically.

Would you like a detailed explanation of how these transformations work?

Here are 5 related questions to expand your understanding:

1. What happens to the graph of $f(x) = x^2$ if you shift it left by 2 units?
2. How does a vertical stretch affect the function $f(x) = x^2$?
3. What is the equation of $f(x) = x^2$ if the graph is reflected over the x-axis?
4. How do you find the vertex of a transformed quadratic function?
5. What is the effect of adding a coefficient in front of $(x - 3)^2$ on the graph?

Tip: When dealing with transformations, horizontal shifts involve changes inside the parentheses (i.e., $(x - h)$), while vertical shifts affect the constant outside (i.e., $+ k$).