Question

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 $g(x) = f(x + 4)$?

Explanation:

Step1: Recall transformation of functions

For a function \( y = f(x) \), the transformation \( y = f(x + h) \) represents a horizontal shift. If \( h>0 \), it is a shift to the left by \( h \) units; if \( h < 0 \), it is a shift to the right by \(|h|\) units.

Step2: Analyze the given transformation

We have \( g(x)=f(x + 4) \), which is of the form \( f(x+h) \) with \( h = 4 \). The original vertex of \( f(x) \) is \( (6,5) \). When we apply the transformation \( f(x+4) \), the \( x \)-coordinate of the vertex will be shifted. Since \( h = 4>0 \), the shift is to the left by 4 units.

Step3: Calculate the new vertex

The original \( x \)-coordinate of the vertex is 6. After shifting left by 4 units, the new \( x \)-coordinate is \( 6-4=2 \). The \( y \)-coordinate remains the same because horizontal shifts do not affect the \( y \)-coordinate of the vertex. So the vertex of \( g(x) \) is \( (2,5) \).

Answer:

The vertex of the function \( g(x)=f(x + 4) \) is \((2,5)\)