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What kind of transformation converts the graph of f(x)=−10∣x+6∣−6f(x) = -10|x + 6| - 6 into the graph of g(x)=−10∣x+5∣−6g(x) = -10|x + 5| - 6?\newlineChoices:\newline(A) translation 11 unit right\newline(B) translation 11 unit up\newline(C) translation 11 unit left\newline(D) translation 11 unit down

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Q. What kind of transformation converts the graph of f(x)=−10∣x+6∣−6f(x) = -10|x + 6| - 6 into the graph of g(x)=−10∣x+5∣−6g(x) = -10|x + 5| - 6?\newlineChoices:\newline(A) translation 11 unit right\newline(B) translation 11 unit up\newline(C) translation 11 unit left\newline(D) translation 11 unit down
  1. Identify Vertex: Identify the vertex of the absolute value function f(x)f(x). The vertex of f(x)=−10∣x+6∣−6f(x) = -10|x + 6| - 6 is at the point where the expression inside the absolute value is zero, which is at x=−6x = -6. The vertex is (−6,−6)(-6, -6).
  2. Identify Vertex: Identify the vertex of the absolute value function g(x)g(x). The vertex of g(x)=−10∣x+5∣−6g(x) = -10|x + 5| - 6 is at the point where the expression inside the absolute value is zero, which is at x=−5x = -5. The vertex is (−5,−6)(-5, -6).
  3. Compare Vertices: Compare the vertices of f(x)f(x) and g(x)g(x) to determine the transformation.\newlineThe vertex of f(x)f(x) is at (−6,−6)(-6, -6) and the vertex of g(x)g(x) is at (−5,−6)(-5, -6). The yy-coordinates are the same, so there is no vertical shift. The xx-coordinate of g(x)g(x) is 11 unit greater than the xx-coordinate of f(x)f(x), indicating a horizontal shift to the right.
  4. Determine Shift: Determine the direction and magnitude of the horizontal shift. Since the xx-coordinate of the vertex of g(x)g(x) is 11 unit greater than the xx-coordinate of the vertex of f(x)f(x), the graph of f(x)f(x) has been translated 11 unit to the right to obtain the graph of g(x)g(x).

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