How is g(x) = (x - 1)^2 related to the graph of f(x) = x^2?

The function g(x) = (x - 1)^2 is related to the graph of f(x) = x^2 through a horizontal shift or translation of the original graph.

1. Starting with the basic function f(x) = x^2, which sketches a parabola with its vertex at the origin (0,0), the graph opens upwards since the coefficient of x^2 is positive.

2. The function g(x) = (x - 1)^2 can be derived from f(x) = x^2 by replacing the variable x in f(x) with (x - 1). This is a transformation of the graph of f(x) that shifts it horizontally.

3. The graph of g(x) = (x - 1)^2 will have the same shape as the graph of f(x) = x^2 since it's still a squared term, but it will be shifted 1 unit to the right. This happens because the term (x - 1) indicates that for each x value on the graph of f(x), you have to subtract 1 to get the corresponding x value on the graph of g(x).

4. As a result, the vertex of the parabola for g(x) = (x - 1)^2 is now at (1,0) instead of the origin.

5. All other features of the parabola remain the same, meaning that it has the same axis of symmetry (now x = 1 instead of x = 0), the same direction of opening (upwards), and the same shape (a U-shaped curve).