To write the function rule for $g(x)$, we can observe the graph and determine how the base function $f(x) = x^2$ has been transformed. The general form for the transformed function is:

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

Where:

• $a$ represents vertical stretching or shrinking (or reflection if negative).
• $h$ represents horizontal translation (shifting left or right).
• $k$ represents vertical translation (shifting up or down).

Step-by-step analysis of the graph:

1. Identify the vertex (h, k): From the graph, we can see that the vertex of the parabola is at $(0, 5)$. This tells us: $h = 0 \quad \text{and} \quad k = 5$

2. Determine the vertical stretch/compression (a): Since the parabola opens upwards and appears to have the same "width" as the base function $f(x) = x^2$, there is no vertical stretch or compression. Therefore: $a = 1$

Final function:

The function rule for $g(x)$ is:

$g(x) = 1(x - 0)^2 + 5 = x^2 + 5$

So the function is:

$g(x) = x^2 + 5$

Let me know if you'd like further details!

Related Questions:

1. How does the value of $a$ affect the graph of a quadratic function?
2. What changes occur when the vertex is shifted horizontally by 2 units?
3. How would the function change if the parabola opened downward?
4. What is the impact of changing $k$ on the graph of the function?
5. How would the equation look if the vertex was at $(2, -3)$?

Tip:

The vertex form of a quadratic function is extremely useful for quickly identifying shifts in the graph and changes to the parabola's shape.