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What kind of transformation converts the graph of f(x)=−4∣x−2∣f(x) = -4|x - 2| into the graph of g(x)=−4∣x−2∣−6g(x) = -4|x - 2| - 6?\newlineChoices:\newline(A) translation 66 units down\newline(B) translation 66 units up\newline(C) translation 66 units left\newline(D) translation 66 units right

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Q. What kind of transformation converts the graph of f(x)=−4∣x−2∣f(x) = -4|x - 2| into the graph of g(x)=−4∣x−2∣−6g(x) = -4|x - 2| - 6?\newlineChoices:\newline(A) translation 66 units down\newline(B) translation 66 units up\newline(C) translation 66 units left\newline(D) translation 66 units right
  1. Identify Vertex: Identify the vertex of the function f(x)=−4∣x−2∣f(x) = -4|x - 2|.\newlineSince it's an absolute value function, the vertex is at x=2x = 2. The y-value is f(2)=−4∣2−2∣=−4∣0∣=0f(2) = -4|2 - 2| = -4|0| = 0.\newlineVertex of f(x)f(x): (2,0)(2, 0)
  2. Identify Vertex: Identify the vertex of the function g(x)=−4∣x−2∣−6g(x) = -4|x - 2| - 6. The vertex is still at x=2x = 2 since the absolute value expression didn't change. The y-value is g(2)=−4∣2−2∣−6=−4∣0∣−6=−6g(2) = -4|2 - 2| - 6 = -4|0| - 6 = -6. Vertex of g(x)g(x): (2,−6)(2, -6)
  3. Determine Transformation: Determine the transformation from f(x)f(x) to g(x)g(x). The xx-coordinate of the vertex didn't change, so there's no horizontal shift. The yy-coordinate went from 00 to −6-6, which is a shift of 66 units down.

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