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What kind of transformation converts the graph of f(x)=9(x+9)2+10f(x) = 9(x + 9)^2 + 10 into the graph of g(x)=9(x+4)2+10g(x) = 9(x + 4)^2 + 10?\newlineChoices:\newline(A) translation 55 units left\newline(B) translation 55 units up\newline(C) translation 55 units down\newline(D) translation 55 units right

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Q. What kind of transformation converts the graph of f(x)=9(x+9)2+10f(x) = 9(x + 9)^2 + 10 into the graph of g(x)=9(x+4)2+10g(x) = 9(x + 4)^2 + 10?\newlineChoices:\newline(A) translation 55 units left\newline(B) translation 55 units up\newline(C) translation 55 units down\newline(D) translation 55 units right
  1. Identify Vertex: Identify the vertex of the function f(x)f(x). The function f(x)=9(x+9)2+10f(x) = 9(x + 9)^2 + 10 is in vertex form, where the vertex is at (−9,10)(-9, 10).
  2. Determine Transformation Type: Identify the vertex of the function g(x)g(x). The function g(x)=9(x+4)2+10g(x) = 9(x + 4)^2 + 10 is also in vertex form, where the vertex is at (−4,10)(-4, 10).
  3. Determine Transformation Direction: Determine the type of transformation.\newlineSince the yy-coordinate of the vertex remains the same (1010) and only the xx-coordinate changes, we are dealing with a horizontal transformation.
  4. Calculate Horizontal Shift: Determine the direction of the transformation.\newlineThe 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 been shifted to the right.
  5. Calculate Horizontal Shift: Determine the direction of the transformation.\newlineThe x-coordinate of the vertex of f(x)f(x) is −9-9, and the x-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 been shifted to the right.Calculate the amount of horizontal shift.\newlineThe difference in the x-coordinates of the vertices is ∣−4−(−9)∣=∣−4+9∣=∣5∣=5|-4 - (-9)| = |-4 + 9| = |5| = 5. Therefore, the graph of f(x)f(x) has been shifted 55 units to the right to become g(x)g(x).

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