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What kind of transformation converts the graph of f(x)=−6∣x−4∣+1f(x) = -6|x - 4| + 1 into the graph of g(x)=−6∣x−4∣+8g(x) = -6|x - 4| + 8?\newlineChoices:\newline(A) translation 77 units left\newline(B) translation 77 units down\newline(C) translation 77 units up\newline(D) translation 77 units right

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Q. What kind of transformation converts the graph of f(x)=−6∣x−4∣+1f(x) = -6|x - 4| + 1 into the graph of g(x)=−6∣x−4∣+8g(x) = -6|x - 4| + 8?\newlineChoices:\newline(A) translation 77 units left\newline(B) translation 77 units down\newline(C) translation 77 units up\newline(D) translation 77 units right
  1. Identify Vertex f(x)f(x): Identify the vertex of the function f(x)=−6∣x−4∣+1f(x) = -6|x - 4| + 1. The vertex of the absolute value function f(x)=−6∣x−4∣+1f(x) = -6|x - 4| + 1 is at the point where the expression inside the absolute value is zero, which is at x=4x = 4. The yy-coordinate of the vertex is the constant term, which is +1+1. Therefore, the vertex of f(x)f(x) is (4,1)(4, 1).
  2. Identify Vertex g(x)g(x): Identify the vertex of the function g(x)=−6∣x−4∣+8g(x) = -6|x - 4| + 8. Similarly, the vertex of the absolute value function g(x)=−6∣x−4∣+8g(x) = -6|x - 4| + 8 is at the point where the expression inside the absolute value is zero, which is at x=4x = 4. The yy-coordinate of the vertex is the constant term, which is +8+8. Therefore, the vertex of g(x)g(x) is (4,8)(4, 8).
  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 change in the yy-coordinate of the vertex from 11 to 88. This is a vertical shift. To find the amount of the shift, subtract the yy-coordinate of the vertex of f(x)f(x) from the yy-coordinate of the vertex of g(x)g(x): g(x)g(x)11. This indicates a translation g(x)g(x)22 units up.

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