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What kind of transformation converts the graph of f(x)=−6(x+2)2+7f(x) = -6(x + 2)^2 + 7 into the graph of g(x)=−6(x+2)2+10g(x) = -6(x + 2)^2 + 10?\newlineChoices:\newline(A) translation 33 units down\newline(B) translation 33 units left\newline(C) translation 33 units right\newline(D) translation 33 units up

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Q. What kind of transformation converts the graph of f(x)=−6(x+2)2+7f(x) = -6(x + 2)^2 + 7 into the graph of g(x)=−6(x+2)2+10g(x) = -6(x + 2)^2 + 10?\newlineChoices:\newline(A) translation 33 units down\newline(B) translation 33 units left\newline(C) translation 33 units right\newline(D) translation 33 units up
  1. Analyze Functions: Analyze the given functions f(x)=−6(x+2)2+7f(x) = -6(x + 2)^2 + 7 and g(x)=−6(x+2)2+10g(x) = -6(x + 2)^2 + 10 to determine the type of transformation.\newlineNotice that the only difference between f(x)f(x) and g(x)g(x) is the constant term at the end of the equation. The (x+2)2(x + 2)^2 term remains unchanged, which means there is no horizontal shift. The coefficient −6-6 in front of the squared term is also unchanged, which means there is no reflection or vertical stretch/compression. The change is solely in the constant term, which indicates a vertical shift.
  2. Determine Vertical Shift: Determine the direction and magnitude of the vertical shift. The constant term in f(x)f(x) is +7+7, and in g(x)g(x) it is +10+10. To go from +7+7 to +10+10, we add 33. This means the graph of f(x)f(x) is shifted up by 33 units to obtain the graph of g(x)g(x).
  3. Match Transformation: Match the transformation with the given choices.\newlineThe graph of f(x)f(x) has been shifted up by 33 units, which corresponds to choice (D) translation 33 units up.

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