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What kind of transformation converts the graph of f(x)=3∣x−3∣−9f(x) = 3|x - 3| - 9 into the graph of g(x)=3∣x+2∣−9g(x) = 3|x + 2| - 9?\newlineChoices:\newline(A) translation 55 units left\newline(B) translation 55 units up\newline(C) translation 55 units down\newline(D) translation 55 units right

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Describe function transformationsright chevron icon

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Q. What kind of transformation converts the graph of f(x)=3∣x−3∣−9f(x) = 3|x - 3| - 9 into the graph of g(x)=3∣x+2∣−9g(x) = 3|x + 2| - 9?\newlineChoices:\newline(A) translation 55 units left\newline(B) translation 55 units up\newline(C) translation 55 units down\newline(D) translation 55 units right
  1. Identify Shape and Orientation: Identify the basic shape and orientation of the graphs for both functions.\newlineBoth functions are in the form of y=a∣x−h∣+ky = a|x - h| + k, which represents a V-shaped graph with its vertex at the point (h,k)(h, k). The coefficient 'aa' determines the opening direction and the steepness of the graph. Since 'aa' is the same for both functions (a=3a = 3), the orientation and steepness of the graphs will be the same.
  2. Determine Vertex of f(x)f(x): Determine the vertex of the original function f(x)f(x). For f(x)=3∣x−3∣−9f(x) = 3|x - 3| - 9, the vertex is at the point where the expression inside the absolute value is zero. This occurs when x−3=0x - 3 = 0, so x=3x = 3. The vertex of f(x)f(x) is therefore at (3,−9)(3, -9).
  3. Determine Vertex of g(x)g(x): Determine the vertex of the transformed function g(x)g(x). For g(x)=3∣x+2∣−9g(x) = 3|x + 2| - 9, the vertex is at the point where the expression inside the absolute value is zero. This occurs when x+2=0x + 2 = 0, so x=−2x = -2. The vertex of g(x)g(x) is therefore at (−2,−9)(-2, -9).
  4. Compare Vertices for Transformation: Compare the vertices of the original and transformed functions to determine the type of transformation.\newlineThe vertex of f(x)f(x) is at (3,−9)(3, -9) and the vertex of g(x)g(x) is at (−2,−9)(-2, -9). The xx-coordinate has changed from 33 to −2-2, which is a shift of 55 units to the left. The yy-coordinate has not changed, so there is no vertical shift.

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