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What kind of transformation converts the graph of f(x)=−7∣x+4∣+7f(x) = -7|x + 4| + 7 into the graph of g(x)=−7∣x−3∣+7g(x) = -7|x - 3| + 7?\newlineChoices:\newline(A) translation 77 units down\newline(B) translation 77 units right\newline(C) translation 77 units left\newline(D) translation 77 units up

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Q. What kind of transformation converts the graph of f(x)=−7∣x+4∣+7f(x) = -7|x + 4| + 7 into the graph of g(x)=−7∣x−3∣+7g(x) = -7|x - 3| + 7?\newlineChoices:\newline(A) translation 77 units down\newline(B) translation 77 units right\newline(C) translation 77 units left\newline(D) translation 77 units up
  1. Identify Vertex of f(x)f(x): Identify the vertex of the function f(x)=−7∣x+4∣+7f(x) = -7|x + 4| + 7. The vertex of the absolute value function f(x)=−7∣x+4∣+7f(x) = -7|x + 4| + 7 occurs where the expression inside the absolute value is zero, i.e., x+4=0x + 4 = 0, which gives x=−4x = -4. The y-coordinate of the vertex is the constant term outside the absolute value, which is +7+7. Therefore, the vertex of f(x)f(x) is (−4,7)(-4, 7).
  2. Identify Vertex of g(x)g(x): Identify the vertex of the function g(x)=−7∣x−3∣+7g(x) = -7|x - 3| + 7. Similarly, the vertex of the absolute value function g(x)=−7∣x−3∣+7g(x) = -7|x - 3| + 7 occurs where the expression inside the absolute value is zero, i.e., x−3=0x - 3 = 0, which gives x=3x = 3. The yy-coordinate of the vertex is the same as in f(x)f(x), which is +7+7. Therefore, the vertex of g(x)g(x) is (3,7)(3, 7).
  3. Determine Transformation: Determine the transformation from f(x)f(x) to g(x)g(x). The transformation from f(x)f(x) to g(x)g(x) involves a horizontal shift from the vertex (−4,7)(-4, 7) to the vertex (3,7)(3, 7). The shift is to the right by 77 units since −4+7=3-4 + 7 = 3.

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