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f(x)=-x^(2)-14 x-48 Which of the following is an equivalent form of the function f in which the maximum value of f appears as a constant or coefficient? Choose 1 answer: (A) f(x)=-(x-7)^(2)+1 (B) f(x)=-(x+7)^(2)+1 (C) f(x)=-(x+6)(x+8) (D) f(x)=-(x-8)(x-6)

f(x)=−x2−14x−48 f(x)=-x^{2}-14 x-48 \newlineWhich of the following is an equivalent form of the function f f in which the maximum value of f f appears as a constant or coefficient?\newlineChoose 11 answer:\newline(A) f(x)=−(x−7)2+1 f(x)=-(x-7)^{2}+1 \newline(B) f(x)=−(x+7)2+1 f(x)=-(x+7)^{2}+1 \newline(C) f(x)=−(x+6)(x+8) f(x)=-(x+6)(x+8) \newline(D) f(x)=−(x−8)(x−6) f(x)=-(x-8)(x-6)

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Q. f(x)=−x2−14x−48 f(x)=-x^{2}-14 x-48 \newlineWhich of the following is an equivalent form of the function f f in which the maximum value of f f appears as a constant or coefficient?\newlineChoose 11 answer:\newline(A) f(x)=−(x−7)2+1 f(x)=-(x-7)^{2}+1 \newline(B) f(x)=−(x+7)2+1 f(x)=-(x+7)^{2}+1 \newline(C) f(x)=−(x+6)(x+8) f(x)=-(x+6)(x+8) \newline(D) f(x)=−(x−8)(x−6) f(x)=-(x-8)(x-6)
  1. Given quadratic function: We are given the quadratic function f(x)=−x2−14x−48f(x) = -x^2 - 14x - 48. To find an equivalent form where the maximum value appears as a constant or coefficient, we need to complete the square.\newlineThe general form of a completed square is a(x−h)2+ka(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.\newlineFirst, we factor out the coefficient of x2x^2, which is −1-1 in this case.\newlinef(x)=−1(x2+14x+48)f(x) = -1(x^2 + 14x + 48)
  2. Factoring out the coefficient: Next, we find the value that completes the square for the expression x2+14xx^2 + 14x. This value is (142)2=49(\frac{14}{2})^2 = 49.\newlineWe add and subtract 4949 inside the parentheses to complete the square.\newlinef(x)=−1(x2+14x+49−49)−48f(x) = -1(x^2 + 14x + 49 - 49) - 48
  3. Completing the square: Now we rewrite the expression by combining the complete square and the constants.\newlinef(x) = −1((x+7)2−49)−48-1((x + 7)^2 - 49) - 48
  4. Rewriting the expression: Simplify the expression by distributing the −1-1 and combining the constants.\newlinef(x)=−(x+7)2+49−48f(x) = -(x + 7)^2 + 49 - 48\newlinef(x)=−(x+7)2+1f(x) = -(x + 7)^2 + 1
  5. Simplifying the expression: We have now expressed f(x)f(x) in vertex form, where the maximum value of the function is the constant term +1+1. The correct answer is the one that matches this form. Comparing with the given options, we see that option (B) f(x)=−(x+7)2+1f(x) = -(x + 7)^2 + 1 is the equivalent form that shows the maximum value as a constant.

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