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 $g(x) = f(x - 2)$?

Explanation:

Step1: Recall function transformation rule

For a function \( y = f(x - h) \), the graph of \( f(x) \) is shifted horizontally by \( h \) units. If \( h>0 \), it is shifted to the right; if \( h < 0 \), it is shifted to the left.

Step2: Analyze the transformation for \( g(x)=f(x - 2) \)

Here, \( h = 2 \). The original vertex of \( f(x) \) is \( (-6,-5) \). When we apply the transformation \( f(x-2) \), we shift the graph of \( f(x) \) 2 units to the right.

To find the new \( x \)-coordinate of the vertex, we add 2 to the original \( x \)-coordinate of the vertex (since shifting right by 2 units). The original \( x \)-coordinate is \( - 6 \), so the new \( x \)-coordinate is \( -6+2=-4 \). The \( y \)-coordinate remains the same because there is no vertical shift, so the new \( y \)-coordinate is \( -5 \).

Answer:

The vertex of \( g(x) \) is \((-4, -5)\)