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What kind of transformation converts the graph of f(x)=−8x2−2f(x) = -8x^2 - 2 into the graph of g(x)=−8(x+8)2−2g(x) = -8(x + 8)^2 - 2?\newlineChoices:\newline(A) translation 88 units left\newline(B) translation 88 units up\newline(C) translation 88 units down\newline(D) translation 88 units right

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Q. What kind of transformation converts the graph of f(x)=−8x2−2f(x) = -8x^2 - 2 into the graph of g(x)=−8(x+8)2−2g(x) = -8(x + 8)^2 - 2?\newlineChoices:\newline(A) translation 88 units left\newline(B) translation 88 units up\newline(C) translation 88 units down\newline(D) translation 88 units right
  1. Analyze Functions: Analyze the given functions.\newlineWe have the original function f(x)=−8x2−2f(x) = -8x^2 - 2 and the transformed function g(x)=−8(x+8)2−2g(x) = -8(x + 8)^2 - 2. We need to determine the type of transformation that occurs between these two functions.
  2. Compare Functions: Compare the two functions.\newlineThe original function f(x)f(x) is in the form of −8x2−2-8x^2 - 2, which is a parabola facing downwards with its vertex at the origin (0,−2)(0, -2). The transformed function g(x)g(x) is in the form of −8(x+8)2−2-8(x + 8)^2 - 2, which is also a parabola facing downwards. The vertex form of a parabola is y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  3. Identify Vertex: Identify the vertex of the transformed function.\newlineThe vertex form of g(x)g(x) is −8(x−(−8))2−2-8(x - (-8))^2 - 2, which means the vertex of g(x)g(x) is at (−8,−2)(-8, -2).
  4. Determine Transformation Type: Determine the type of transformation.\newlineThe vertex of f(x)f(x) is at (0,−2)(0, -2) and the vertex of g(x)g(x) is at (−8,−2)(-8, -2). The yy-coordinate of the vertex has not changed, so there is no vertical translation. The xx-coordinate of the vertex has changed from 00 to −8-8, which indicates a horizontal translation.
  5. Determine Translation Magnitude: Determine the direction and magnitude of the horizontal translation.\newlineThe xx-coordinate of the vertex has moved from 00 to −8-8, which means the graph has shifted 88 units to the left.

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