A quadratic function $$ y=f(x) $$ is plotted on a graph and the vertex of the resulting parabola is $$ (4,-5) $$. What is the vertex of the function defined as $$ g(x)=f(x+2)+2 $$ ?

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The vertex of $$ g(x) = f(x+2) + 2 $$ is $$ (2, -3) $$. To find the vertex of the function $$ g(x) = f(x+2) + 2 $$, we start with the vertex of the original function $$ f(x) $$, which is given as $$ (4, -5) $$. The vertex of a quadratic function $$ f(x) $$ is represented as $$ (h, k) $$, where $$ h $$ is the x-coordinate and $$ k $$ is the y-coordinate. In this case, we have: $$ h = 4, \quad k = -5 $$ Now, we need to analyze the transformation applied to $$ f(x) $$ to obtain $$ g(x) $$. 1. **Horizontal Shift**: The term $$ f(x+2) $$ indicates a horizontal shift to the left by 2 units. This means that the x-coordinate of the vertex will change from $$ h = 4 $$ to: $$ h' = 4 - 2 = 2 $$ 2. **Vertical Shift**: The term $$ +2 $$ indicates a vertical shift upwards by 2 units. This means that the y-coordinate of the vertex will change from $$ k = -5 $$ to: $$ k' = -5 + 2 = -3 $$ Combining these transformations, the new vertex of the function $$ g(x) $$ is: $$ (h', k') = (2, -3) $$ Thus, the vertex of the function $$ g(x) = f(x+2) + 2 $$ is $$ \boxed{(2, -3)} $$

A quadratic function \( y=f(x) \) is plotted on a graph and the vertex of the resulting parabola is \( (4,-5) \). What is the vertex of the function defined as \( g(x)=f(x+2)+2 \) ?

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A quadratic function \( y=f(x) \) is plotted on a graph and the vertex of the resulting parabola is \( (4,-5) \). What is the vertex of the function defined as \( g(x)=f(x+2)+2 \) ?

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