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What kind of transformation converts the graph of f(x)=−6(x+9)2−5f(x) = -6(x + 9)^2 - 5 into the graph of g(x)=−6(x+4)2−5g(x) = -6(x + 4)^2 - 5?\newlineChoices:\newline(A) translation 55 units right\newline(B) translation 55 units down\newline(C) translation 55 units left\newline(D) translation 55 units up

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Q. What kind of transformation converts the graph of f(x)=−6(x+9)2−5f(x) = -6(x + 9)^2 - 5 into the graph of g(x)=−6(x+4)2−5g(x) = -6(x + 4)^2 - 5?\newlineChoices:\newline(A) translation 55 units right\newline(B) translation 55 units down\newline(C) translation 55 units left\newline(D) translation 55 units up
  1. Identify Vertex f(x)f(x): Identify the vertex of the function f(x)f(x). The function f(x)=−6(x+9)2−5f(x) = -6(x + 9)^2 - 5 is in vertex form, where the vertex is at (−h,k)(-h, k). For f(x)f(x), the vertex is at (−9,−5)(-9, -5).
  2. Identify Vertex g(x)g(x): Identify the vertex of the function g(x)g(x). The function g(x)=−6(x+4)2−5g(x) = -6(x + 4)^2 - 5 is also in vertex form. For g(x)g(x), the vertex is at (−4,−5)(-4, -5).
  3. Type of Transformation: Determine the type of transformation.\newlineThe yy-coordinates of the vertices of f(x)f(x) and g(x)g(x) are the same, which means there is no vertical shift. The xx-coordinates have changed, indicating a horizontal shift.
  4. Direction of Shift: Determine the direction of the horizontal shift. The xx-coordinate of the vertex of f(x)f(x) is −9-9, and the xx-coordinate of the vertex of g(x)g(x) is −4-4. Since −4-4 is to the right of −9-9 on the number line, the graph has shifted to the right.
  5. Magnitude of Shift: Calculate the magnitude of the horizontal shift. The difference in the x-coordinates of the vertices is ∣−9−(−4)∣=∣−9+4∣=∣5∣=5|-9 - (-4)| = |-9 + 4| = |5| = 5. Therefore, the graph of f(x)f(x) has shifted 55 units to the right to become the graph of g(x)g(x).

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