Question

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

Explanation:

Step1: Recall vertical shift rule

For a function \( y = f(x) \), the transformation \( g(x)=f(x)+k \) (where \( k \) is a constant) represents a vertical shift. If \( k>0 \), it is a shift up by \( k \) units; if \( k < 0 \), it is a shift down by \( |k| \) units. The \( x \)-coordinate of the vertex remains the same, and the \( y \)-coordinate is shifted by \( k \) units.

Step2: Apply the rule to the given vertex

The vertex of \( f(x) \) is \( (6, - 6) \). For \( g(x)=f(x)+2 \), we are shifting the graph of \( f(x) \) up by 2 units. So the \( x \)-coordinate of the vertex of \( g(x) \) is the same as that of \( f(x) \), which is 6. The \( y \)-coordinate is \( -6 + 2=-4 \).

Answer:

The vertex of \( g(x) \) is \( (6, - 4) \)