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What kind of transformation converts the graph of f(x)=9∣x+6∣−4f(x) = 9|x + 6| - 4 into the graph of g(x)=9∣x+6∣−9g(x) = 9|x + 6| - 9?\newlineChoices:\newline(A) translation 55 units up\newline(B) translation 55 units left\newline(C) translation 55 units down\newline(D) translation 55 units right

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Q. What kind of transformation converts the graph of f(x)=9∣x+6∣−4f(x) = 9|x + 6| - 4 into the graph of g(x)=9∣x+6∣−9g(x) = 9|x + 6| - 9?\newlineChoices:\newline(A) translation 55 units up\newline(B) translation 55 units left\newline(C) translation 55 units down\newline(D) translation 55 units right
  1. Identify Transformation: Identify the transformation between f(x)f(x) and g(x)g(x) by comparing the two functions.\newlineThe function f(x)=9∣x+6∣−4f(x) = 9|x + 6| - 4 is being transformed into g(x)=9∣x+6∣−9g(x) = 9|x + 6| - 9. The only difference between these two functions is the constant term at the end of the expression. Since the constant term in f(x)f(x) is −4-4 and in g(x)g(x) it is −9-9, the transformation involves a vertical shift.
  2. Determine Vertical Shift: Determine the direction and magnitude of the vertical shift. The constant term in f(x)f(x) has decreased by 55 units to become the constant term in g(x)g(x) (from −4-4 to −9-9). This indicates a vertical shift downward by 55 units.
  3. Match Transformation: Match the transformation to the given choices.\newlineThe transformation is a translation 55 units down, which corresponds to choice (C)(C).

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