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What kind of transformation converts the graph of f(x)=−7(x−2)2−4f(x) = -7(x - 2)^2 - 4 into the graph of g(x)=−7(x+5)2−4g(x) = -7(x + 5)^2 - 4?\newlineChoices:\newline(A) translation 77 units up\newline(B) translation 77 units left\newline(C) translation 77 units down\newline(D) translation 77 units right

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Q. What kind of transformation converts the graph of f(x)=−7(x−2)2−4f(x) = -7(x - 2)^2 - 4 into the graph of g(x)=−7(x+5)2−4g(x) = -7(x + 5)^2 - 4?\newlineChoices:\newline(A) translation 77 units up\newline(B) translation 77 units left\newline(C) translation 77 units down\newline(D) translation 77 units right
  1. Identify Vertex: Identify the vertex of the function f(x)f(x). Compare f(x)=−7(x−2)2−4f(x) = -7(x - 2)^2 - 4 with the vertex form of a parabola. Vertex of f(x)f(x): (2,−4)(2, -4)
  2. Compare Functions: Identify the vertex of the transformed function g(x)g(x). Compare g(x)=−7(x+5)2−4g(x) = -7(x + 5)^2 - 4 with the vertex form of a parabola. Vertex of g(x)g(x): (−5,−4)(-5, -4)
  3. Determine Transformation Type: Determine the type of transformation.\newlineThe yy-coordinates of the vertices of f(x)f(x) and g(x)g(x) are the same, so there is no vertical shift.\newlineThe xx-coordinates of the vertices are 22 and −5-5 respectively, indicating a horizontal shift.
  4. Determine Shift Direction: Determine the direction of the horizontal shift. The xx-coordinate of the vertex of f(x)f(x) is 22, and the xx-coordinate of the vertex of g(x)g(x) is −5-5. Since −5-5 is to the left of 22 on the number line, f(x)f(x) shifts to the left to become g(x)g(x).
  5. Calculate Shift Magnitude: Calculate the magnitude of the horizontal shift.\newlineThe difference in x-coordinates of the vertices is ∣2−(−5)∣=∣2+5∣=7|2 - (-5)| = |2 + 5| = 7.\newlineThe graph of f(x)f(x) shifts 77 units to the left.

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