To find the vertex of the function $g(x) = f(x+2) + 4$, we need to understand how the transformation affects the vertex of the original function $f(x)$.

Given:

• The original function $f(x)$ has a vertex at $(4, -6)$.
• The transformed function is $g(x) = f(x+2) + 4$.

Step 1: Horizontal Shift

The term $x + 2$ inside the function $f(x+2)$ represents a horizontal shift to the left by 2 units.

Therefore, the vertex of $f(x)$, originally at $(4, -6)$, moves to the point $(4 - 2, -6) = (2, -6)$.

Step 2: Vertical Shift

The term $+4$ outside the function $f(x+2)$ represents a vertical shift upwards by 4 units.

So, the vertex at $(2, -6)$ will move up by 4 units, resulting in a new vertex at $(2, -6 + 4) = (2, -2)$.

Final Answer:

The vertex of the function $g(x) = f(x+2) + 4$ is $\boxed{(2, -2)}$.

Would you like more details or have any further questions?

Here are five related questions:

1. How does a horizontal shift affect the graph of a quadratic function?
2. What is the effect of a vertical shift on the vertex of a quadratic function?
3. Can you find the vertex of a function if it undergoes both horizontal and vertical shifts?
4. How does the vertex form of a quadratic function help in identifying shifts?
5. What would be the vertex if the transformation was $g(x) = f(x-3) - 5$?

Tip: When working with function transformations, always apply horizontal shifts before vertical ones.