Question

a quadratic function $y = f(x)$ is plotted on a graph and the vertex of the resulting parabola is $(5, 6)$. what is the vertex of the function defined as $g(x) = f(x) - 4$?

Explanation:

Step1: Recall vertical shift rule

For a function \( y = f(x) \), the transformation \( g(x)=f(x)-k \) (where \( k>0 \)) shifts the graph of \( f(x) \) downward by \( k \) units. This shift affects the \( y \)-coordinate of the vertex, while the \( x \)-coordinate remains the same.

Step2: Apply the rule to the vertex

The vertex of \( f(x) \) is \( (5, 6) \). For \( g(x)=f(x)-4 \), we are shifting the graph of \( f(x) \) down by 4 units. So the \( x \)-coordinate of the vertex of \( g(x) \) remains 5, and the \( y \)-coordinate is \( 6 - 4=2 \).

Answer:

The vertex of \( g(x) \) is \( (5, 2) \)