The problem involves the transformation of the quadratic function $f(x)$ to a new function $g(x) = f(x+3)$. To determine the vertex of $g(x)$, we can use the transformation properties of parabolas.

The vertex of the original quadratic function $f(x)$ is at $(-3, 3)$. When you replace $x$ with $x+3$ in the function, the effect is a horizontal shift to the left by 3 units.

Here's how the transformation works:

• The general rule is that replacing $x$ with $x + h$ in a function shifts the graph horizontally by $h$ units.
• In this case, $g(x) = f(x+3)$, so this shifts the graph 3 units to the left.

Since the vertex of $f(x)$ is originally at $(-3, 3)$, shifting it 3 units to the left moves it to $(-6, 3)$.

Therefore, the vertex of the function $g(x) = f(x+3)$ is:

$(-6, 3)$

Would you like a more detailed explanation, or do you have any additional questions?

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Here are 5 related questions to explore further:

1. What is the equation of the parabola if the function $f(x)$ is given explicitly?
2. How do vertical shifts affect the vertex of a quadratic function?
3. What other transformations could we apply to the function, such as reflections or stretches?
4. How does the form $f(x) = a(x-h)^2 + k$ help in identifying the vertex easily?
5. What happens if we combine horizontal and vertical shifts in the same transformation?

Tip: Remember, horizontal transformations affect the $x$-coordinate, while vertical transformations affect the $y$-coordinate of the vertex!