Question

the graph of $f(x) = x^2$ is translated to form $g(x) = (x - 5)^2 + 1$. which graph represents $g(x)$?

Explanation:

Step1: Recall Vertex Form of Parabola

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( f(x)=x^2 \), the vertex is \((0, 0)\). For \( g(x)=(x - 5)^2+1 \), comparing with \( y = a(x - h)^2 + k \), we have \( h = 5 \) and \( k = 1 \), so the vertex of \( g(x) \) is \((5, 1)\).

Step2: Analyze the Direction and Shape

Since the coefficient of \((x - 5)^2\) is \( 1 \) (positive), the parabola opens upwards, same as \( f(x)=x^2 \). So we need to find the graph with vertex at \((5, 1)\) and opening upwards.

Answer:

The graph of \( g(x) \) is a parabola opening upwards with vertex at \((5, 1)\). (Among the given options, the one with vertex \((5, 1)\) and upward opening is the correct representation of \( g(x) \).)