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What kind of transformation converts the graph of f(x)=−9(x−9)2−9f(x) = -9(x - 9)^2 - 9 into the graph of g(x)=−9(x+1)2−9g(x) = -9(x + 1)^2 - 9?\newlineChoices:\newline(A) translation 1010 units left\newline(B) translation 1010 units up\newline(C) translation 1010 units right\newline(D) translation 1010 units down

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Q. What kind of transformation converts the graph of f(x)=−9(x−9)2−9f(x) = -9(x - 9)^2 - 9 into the graph of g(x)=−9(x+1)2−9g(x) = -9(x + 1)^2 - 9?\newlineChoices:\newline(A) translation 1010 units left\newline(B) translation 1010 units up\newline(C) translation 1010 units right\newline(D) translation 1010 units down
  1. Find Vertex of f(x)f(x): Analyze the given function f(x)=−9(x−9)2−9f(x) = -9(x - 9)^2 - 9 to find its vertex.\newlineThe vertex form of a quadratic function is f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.\newlineFor f(x)f(x), the vertex is at (h,k)=(9,−9)(h, k) = (9, -9).
  2. Find Vertex of g(x)g(x): Analyze the transformed function g(x)=−9(x+1)2−9g(x) = -9(x + 1)^2 - 9 to find its vertex.\newlineUsing the vertex form, the vertex of g(x)g(x) is at (h,k)=(−1,−9)(h, k) = (-1, -9).
  3. Compare Vertices for Transformation: Determine the type of transformation by comparing the vertices of f(x)f(x) and g(x)g(x). The vertex of f(x)f(x) is (9,−9)(9, -9), and the vertex of g(x)g(x) is (−1,−9)(-1, -9). Since the yy-coordinates of the vertices are the same, there is no vertical shift. The xx-coordinate of the vertex of g(x)g(x) is 1010 units to the left of the xx-coordinate of the vertex of f(x)f(x).
  4. Calculate Horizontal Shift: Calculate the exact horizontal shift from the vertex of f(x)f(x) to the vertex of g(x)g(x). The shift is the difference in the xx-coordinates of the vertices: 9−(−1)=9+1=109 - (-1) = 9 + 1 = 10. The graph of f(x)f(x) is shifted 1010 units to the left to obtain the graph of g(x)g(x).

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