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Select the correct answer. If the parent function is f(x) = x2 and its transformed function is g(x) = 2(x − 1)2 + 2, how is the graph of g(x) transformed from the graph of its parent function? A. stretch, shift along the y–axis, and shift along the x–axis B. shift along the x–axis and the y–axis C. reflection only D. stretch along the y–axis and reflection through the origin

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Answer:

A) stretch, shift along the y–axis, and shift along the x–axis.

Step-by-step explanation:

Given the parent quadratic function, f(x) = x², and the transformed function in vertex form, g(x) = 2(x - 1)² + 2:

In the vertex form, g(x) = a(x - h)² + k

where:

(h, k) = vertex

a = makes the parent function wider (0 < a < 1) or narrower (a > 1).

h = determines the horizontal translation of the parent graph:

→ Horizontal translation of h units to the right: g(x) = f(x - h), where h > 0

→ Horizontal translation of |h| units to the left: g(x) = f(x - h), where h < 0.

k = determines the vertical translation of the parent graph.

Given these definitions, it is evident that the transformed function is narrower (vertical stretch by a factor of a = 2); shifted along the y-axis (with k = 2), and horizontally shifted along the x-axis given h = 1.

Therefore, the correct answer is Option A): stretch, shift along the y–axis, and shift along the x–axis.

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