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What kind of transformation converts the graph of f(x)=−10∣x∣+3f(x) = -10|x| + 3 into the graph of g(x)=−10∣x−6∣+3g(x) = -10|x - 6| + 3?\newlineChoices:\newline(A) translation 66 units right\newline(B) translation 66 units down\newline(C) translation 66 units up\newline(D) translation 66 units left

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Q. What kind of transformation converts the graph of f(x)=−10∣x∣+3f(x) = -10|x| + 3 into the graph of g(x)=−10∣x−6∣+3g(x) = -10|x - 6| + 3?\newlineChoices:\newline(A) translation 66 units right\newline(B) translation 66 units down\newline(C) translation 66 units up\newline(D) translation 66 units left
  1. Identify Shape and Vertex: Identify the basic shape and vertex of the function f(x)=−10∣x∣+3f(x) = -10|x| + 3.\newlineThe function f(x)=−10∣x∣+3f(x) = -10|x| + 3 is a V-shaped graph with its vertex at the origin (0,3)(0, 3) because the absolute value function ∣x∣|x| has a vertex at (0,0)(0, 0) and the +3+3 shifts it up by 33 units.
  2. Identify Shape and Vertex: Identify the basic shape and vertex of the function g(x)=−10∣x−6∣+3g(x) = -10|x - 6| + 3.\newlineThe function g(x)=−10∣x−6∣+3g(x) = -10|x - 6| + 3 is also a V-shaped graph. The term ∣x−6∣|x - 6| indicates a horizontal shift of the absolute value function. Since it is ∣x−6∣|x - 6|, the vertex is shifted 66 units to the right, making the new vertex (6,3)(6, 3).
  3. Determine Transformation: Determine the type of transformation from f(x)f(x) to g(x)g(x). The transformation involves a horizontal shift since the vertex of g(x)g(x) has moved from (0,3)(0, 3) to (6,3)(6, 3). The yy-coordinate of the vertex has not changed, so there is no vertical shift. The xx-coordinate has increased by 66, which means the graph has been translated 66 units to the right.
  4. Match Transformation: Match the transformation to the given choices.\newlineThe transformation is a translation 66 units to the right, which corresponds to choice (A)(A).

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