A quadratic function y, equals, f, of, xy=f(x) is plotted on a graph and the vertex of the resulting parabola is left parenthesis, 6, comma, 5, right parenthesis(6,5). What is the vertex of the function defined as g, of, x, equals, f, of, x, plus, 2g(x)=f(x+2)

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).

A quadratic function y = f(x) is plotted on a graph and the vertex of the resulting parabola is (6, 5). What is the vertex of the function defined as g(x) = f(x + 2)?

Learn how to determine the new vertex of a quadratic function after a horizontal shift. This problem covers essential algebraic concepts, including quadratic functions and graph transformations. Suitable for students in Grades 8-10.

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