Given the quadratic function $y = f(x)$ with a vertex at $(6, 5)$, we are asked to find the vertex of the transformed function $g(x) = f(x + 2)$.

Understanding the Transformation

The function $g(x) = f(x + 2)$ represents a horizontal shift of the function $f(x)$. Specifically:

• $f(x + 2)$ shifts the graph of $f(x)$ to the left by 2 units.

Determining the New Vertex

The original vertex of $f(x)$ is at $(6, 5)$. After applying the transformation $f(x + 2)$:

• The x-coordinate of the vertex moves from $6$ to $6 - 2 = 4$.
• The y-coordinate remains unchanged at 5.

Therefore, the vertex of the function $g(x) = f(x + 2)$ is $\mathbf{(4, 5)}$.

Would you like further details on how transformations affect quadratic functions? Here are some related questions you might find helpful:

1. How do vertical shifts affect the vertex of a quadratic function?
2. What happens to the vertex if the function is reflected across the x-axis?
3. How would the vertex change if the function was $g(x) = f(x - 2)$ instead?
4. Can you determine the axis of symmetry for the transformed function $g(x)$?
5. How do horizontal stretches and compressions affect the vertex of a quadratic function?

Tip: When dealing with horizontal shifts, always remember that adding a constant inside the function moves the graph in the opposite direction (i.e., $f(x + h)$ shifts left, and $f(x - h)$ shifts right).