the graph of $f(x)=x^{2}$ is translated to form $g(x)=(x - 5)^{2}+1$. which graph represents $g(x)$?
the graph of $f(x)=x^{2}$ is translated to form $g(x)=(x - 5)^{2}+1$. which graph represents $g(x)$?
Answer
Answer:
The graph of (g(x)=(x - 5)^2+1) is the graph of (f(x)=x^2) translated 5 units to the right and 1 unit up. The vertex of (f(x)=x^2) is ((0,0)), and the vertex of (g(x)=(x - 5)^2+1) is ((5,1)). So the graph with vertex at ((5,1)) and the same parabolic - shape as (y = x^2) represents (g(x)).
Explanation:
Step1: Identify the vertex - form of a quadratic function
The general vertex - form of a quadratic function is (y=a(x - h)^2+k), where ((h,k)) is the vertex of the parabola. For (f(x)=x^2), (a = 1), (h = 0), and (k = 0), so the vertex is ((0,0)).
Step2: Analyze the transformation for (g(x))
For (g(x)=(x - 5)^2+1), we have (a = 1), (h = 5), and (k = 1). The value of (h) represents a horizontal shift. Since (h = 5\gt0), the graph is shifted 5 units to the right. The value of (k) represents a vertical shift. Since (k = 1\gt0), the graph is shifted 1 unit up. So the vertex of (g(x)) is ((5,1)).