The graph provided shows a parabola, which is a transformation of the basic quadratic function $f(x) = x^2$. Based on the image, I will determine the values of $a$, $h$, and $k$ to write the function $g(x)$ in the form:

$g(x) = a(x - h)^2 + k$

Observations from the graph:

1. Vertex: The vertex of the parabola is located at $(4, 2)$, meaning the graph has been translated 4 units to the right and 2 units up. Therefore, $h = 4$ and $k = 2$.

2. Shape and Orientation: The parabola opens upwards, and based on its width, there doesn't appear to be any vertical stretching or compression compared to $f(x) = x^2$. Thus, $a = 1$.

Function Rule:

Given these transformations, the equation for $g(x)$ is:

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

This is the rule for the function $g(x)$, describing the parabola in the graph.

Would you like further details or have any other questions?

Here are 5 follow-up questions for exploration:

1. How would the function change if the parabola were reflected across the x-axis?
2. What happens to the graph if the value of $a$ is greater than 1?
3. How would you describe the effect of $h$ on the graph of a parabola?
4. If $k$ were negative, how would that affect the graph?
5. Can you identify the domain and range of the function $g(x)$?

Tip: When analyzing the transformation of a function, always start by identifying the vertex to help you determine the shifts in the graph.