choose the graph of the function f(x)=2(x - 1)^2 and its parent function.

Explanation:

Step1: Identify the parent - function

The parent function of (f(x)=2(x - 1)^2) is (y = x^2), which is a parabola with vertex at the origin ((0,0)) and opens upward.

Step2: Analyze the transformation of (f(x)=2(x - 1)^2)

For the function (y=a(x - h)^2+k), here (a = 2), (h = 1), and (k = 0). The factor (a=2) causes a vertical stretch by a factor of 2, and the ((x - 1)) term causes a horizontal shift 1 unit to the right. The vertex of (f(x)=2(x - 1)^2) is at ((1,0)).

Step3: Check the graphs

We look for a graph where the blue graph ((y = x^2)) has vertex at ((0,0)) and the red graph ((f(x))) has vertex at ((1,0)) and is vertically stretched compared to (y = x^2).

Answer:

The graph where the parabola (y = x^2) (blue) has vertex at ((0,0)) and the parabola (f(x)=2(x - 1)^2) (red) has vertex at ((1,0)) and is vertically stretched compared to (y = x^2) (usually the third - from - the - top graph if we assume a standard order of presentation). Since no specific identifiers for the graphs are given, we describe the correct graph characteristics as above.