To determine the function rule for $g(x)$, which is a transformation of the parent function $f(x) = x$, given the vertex at $(0, -3)$, let’s analyze the situation.

The form of the function is:

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

Step 1: Identify the values of $h$ and $k$

• The vertex is at $(h, k)$.
• From the given vertex $(0, -3)$: $h = 0 \quad \text{and} \quad k = -3$

Step 2: Determine the value of $a$

• For a linear function like $g(x)$, the slope of $a$ remains consistent as it corresponds to the slope of $f(x) = x$, which is $1$ unless otherwise stated.

Thus, the slope $a = 1$.

Step 3: Write the equation

Substituting $a = 1$, $h = 0$, and $k = -3$ into the formula: $g(x) = 1(x - 0) + (-3)$

Simplify: $g(x) = x - 3$

Final Answer:

$g(x) = x - 3$

Would you like further details or have any questions about the process? Here are five related questions for exploration:

1. How would the function change if the vertex moved to $(2, -3)$?
2. What if the slope $a$ was $2$ instead of $1$?
3. Can you rewrite this function in standard form $y = mx + b$?
4. How do translations affect the domain and range of a function?
5. How does the transformation relate graphically to the original $f(x) = x$?

Tip: The parameter $k$ in a function directly affects vertical translations, moving the graph up or down depending on its sign.