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What kind of transformation converts the graph of f(x)=9∣x+6∣−9f(x) = 9|x + 6| - 9 into the graph of g(x)=9∣x+6∣+1g(x) = 9|x + 6| + 1?\newlineChoices:\newline(A) translation 1010 units right\newline(B) translation 1010 units up\newline(C) translation 1010 units down\newline(D) translation 1010 units left

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Q. What kind of transformation converts the graph of f(x)=9∣x+6∣−9f(x) = 9|x + 6| - 9 into the graph of g(x)=9∣x+6∣+1g(x) = 9|x + 6| + 1?\newlineChoices:\newline(A) translation 1010 units right\newline(B) translation 1010 units up\newline(C) translation 1010 units down\newline(D) translation 1010 units left
  1. Analyze Functions: Analyze the given functions.\newlineWe have f(x)=9∣x+6∣−9f(x) = 9|x + 6| - 9 and g(x)=9∣x+6∣+1g(x) = 9|x + 6| + 1.\newlineCompare the two functions to determine the type of transformation.
  2. Identify Difference: Identify the change in the functions.\newlineThe only difference between f(x)f(x) and g(x)g(x) is the constant term at the end of the equation.\newlinef(x)f(x) has −9-9, and g(x)g(x) has +1+1.
  3. Direction of Transformation: Determine the direction of the transformation. Since the change is in the constant term, this indicates a vertical shift.
  4. Calculate Shift Amount: Calculate the amount of vertical shift.\newlineThe change from −9-9 to +1+1 is an increase of 1010 units.\newlineTo find the shift, we calculate 1−(−9)=1+9=101 - (-9) = 1 + 9 = 10.
  5. Type of Vertical Shift: Determine the type of vertical shift.\newlineSince the constant term increased by 1010, the graph of f(x)f(x) is shifted 1010 units up to become g(x)g(x).

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