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What kind of transformation converts the graph of f(x)=8∣x−2∣−2f(x) = 8|x - 2| - 2 into the graph of g(x)=8∣x+4∣−2g(x) = 8|x + 4| - 2?\newlineChoices:\newline(A) translation 66 units right\newline(B) translation 66 units up\newline(C) translation 66 units left\newline(D) translation 66 units down

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Q. What kind of transformation converts the graph of f(x)=8∣x−2∣−2f(x) = 8|x - 2| - 2 into the graph of g(x)=8∣x+4∣−2g(x) = 8|x + 4| - 2?\newlineChoices:\newline(A) translation 66 units right\newline(B) translation 66 units up\newline(C) translation 66 units left\newline(D) translation 66 units down
  1. Identify Vertex of f(x)f(x): Let's identify the vertex of the absolute value function f(x)=8∣x−2∣−2f(x) = 8|x - 2| - 2. The vertex of f(x)f(x) is at the point where the expression inside the absolute value is zero, which is x=2x = 2. The y-coordinate of the vertex is the value of the function when x=2x = 2, which is f(2)=8∣2−2∣−2=8∣0∣−2=0−2=−2f(2) = 8|2 - 2| - 2 = 8|0| - 2 = 0 - 2 = -2. So the vertex of f(x)f(x) is (2,−2)(2, -2).
  2. Identify Vertex of g(x)g(x): Now let's identify the vertex of the absolute value function g(x)=8∣x+4∣−2g(x) = 8|x + 4| - 2. The vertex of g(x)g(x) is at the point where the expression inside the absolute value is zero, which is x=−4x = -4. The y-coordinate of the vertex is the value of the function when x=−4x = -4, which is g(−4)=8∣−4+4∣−2=8∣0∣−2=0−2=−2g(-4) = 8|-4 + 4| - 2 = 8|0| - 2 = 0 - 2 = -2. So the vertex of g(x)g(x) is (−4,−2)(-4, -2).
  3. Determine Transformation: To determine the transformation, we compare the vertices of f(x)f(x) and g(x)g(x). The vertex of f(x)f(x) is (2,−2)(2, -2) and the vertex of g(x)g(x) is (−4,−2)(-4, -2). The xx-coordinate of the vertex has moved from 22 to −4-4, which is a shift of 66 units to the left. The g(x)g(x)00-coordinate has not changed, so there is no vertical shift.

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