Question

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in the xy-plane, the graph of the quadratic function f is a parabola with vertex (2, 3). the function g is defined by g(x) = f(x) + 2, and the graph of g has vertex (h, k). what is the value of h?
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Explanation:

Step1: Recall transformation of functions

When we have a function \( g(x)=f(x)+c \) (where \( c \) is a constant), this represents a vertical shift of the graph of \( f(x) \). A vertical shift does not affect the \( x \)-coordinate of the vertex (or any point's \( x \)-coordinate), only the \( y \)-coordinate.

Step2: Determine the vertex of \( g(x) \)

The vertex of \( f(x) \) is \( (2, 3) \). Since \( g(x)=f(x)+2 \) is a vertical shift (up by 2 units), the \( x \)-coordinate of the vertex of \( g(x) \) will be the same as that of \( f(x) \), and the \( y \)-coordinate will be \( 3 + 2=5 \). So the vertex of \( g(x) \) is \( (h,k)=(2,5) \), which means \( h = 2 \).

Answer:

2