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What kind of transformation converts the graph of f(x)=2∣x−5∣+1f(x) = 2|x - 5| + 1 into the graph of g(x)=2∣x+5∣+1g(x) = 2|x + 5| + 1?\newlineChoices:\newline(A) translation 1010 units down\newline(B) translation 1010 units up\newline(C) translation 1010 units left\newline(D) translation 1010 units right

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Q. What kind of transformation converts the graph of f(x)=2∣x−5∣+1f(x) = 2|x - 5| + 1 into the graph of g(x)=2∣x+5∣+1g(x) = 2|x + 5| + 1?\newlineChoices:\newline(A) translation 1010 units down\newline(B) translation 1010 units up\newline(C) translation 1010 units left\newline(D) translation 1010 units right
  1. Identify Vertex of f(x)f(x): Identify the vertex of the function f(x)=2∣x−5∣+1f(x) = 2|x - 5| + 1. The vertex of the absolute value function f(x)=2∣x−5∣+1f(x) = 2|x - 5| + 1 is at the point where the expression inside the absolute value is zero. This occurs at x=5x = 5. The yy-coordinate of the vertex is the value of the function when x=5x = 5, which is f(5)=2∣5−5∣+1=2∣0∣+1=2(0)+1=1f(5) = 2|5 - 5| + 1 = 2|0| + 1 = 2(0) + 1 = 1. Therefore, the vertex of f(x)f(x) is (5,1)(5, 1).
  2. Identify Vertex of g(x)g(x): Identify the vertex of the function g(x)=2∣x+5∣+1g(x) = 2|x + 5| + 1. Similarly, the vertex of the absolute value function g(x)=2∣x+5∣+1g(x) = 2|x + 5| + 1 is at the point where the expression inside the absolute value is zero. This occurs at x=−5x = -5. The yy-coordinate of the vertex is the value of the function when x=−5x = -5, which is g(−5)=2∣−5+5∣+1=2∣0∣+1=2(0)+1=1g(-5) = 2|-5 + 5| + 1 = 2|0| + 1 = 2(0) + 1 = 1. Therefore, the vertex of g(x)g(x) is (−5,1)(-5, 1).
  3. Determine Transformation Type: Determine the type of transformation from f(x)f(x) to g(x)g(x). The vertex of f(x)f(x) is (5,1)(5, 1) and the vertex of g(x)g(x) is (−5,1)(-5, 1). The transformation from f(x)f(x) to g(x)g(x) involves a horizontal shift, since the yy-coordinates of the vertices are the same, but the xx-coordinates are different. The shift is from g(x)g(x)00 to g(x)g(x)11, which is a shift of g(x)g(x)22 units to the left.

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