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f(x)=-(x+1)(x+7) The function represents a parabola in the xy-plane. Which of the following is an equivalent form of f in which the y-intercept of the graph of f appears as a constant or coefficient? Choose 1 answer: (A) f(x)=-(x+4)^(2)+9 (B) f(x)=-(x+4)^(2)-9 (c) f(x)=-x^(2)+8x+7 (D) f(x)=-x^(2)-8x-7

f(x)=−(x+1)(x+7) f(x)=-(x+1)(x+7) \newlineThe function represents a parabola in the xy x y -plane. Which of the following is an equivalent form of f f in which the y y -intercept of the graph of f f appears as a constant or coefficient?\newlineChoose 11 answer:\newline(A) f(x)=−(x+4)2+9 f(x)=-(x+4)^{2}+9 \newline(B) f(x)=−(x+4)2−9 f(x)=-(x+4)^{2}-9 \newline(C) f(x)=−x2+8x+7 f(x)=-x^{2}+8 x+7 \newline(D) f(x)=−x2−8x−7 f(x)=-x^{2}-8 x-7

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Full solution

Q. f(x)=−(x+1)(x+7) f(x)=-(x+1)(x+7) \newlineThe function represents a parabola in the xy x y -plane. Which of the following is an equivalent form of f f in which the y y -intercept of the graph of f f appears as a constant or coefficient?\newlineChoose 11 answer:\newline(A) f(x)=−(x+4)2+9 f(x)=-(x+4)^{2}+9 \newline(B) f(x)=−(x+4)2−9 f(x)=-(x+4)^{2}-9 \newline(C) f(x)=−x2+8x+7 f(x)=-x^{2}+8 x+7 \newline(D) f(x)=−x2−8x−7 f(x)=-x^{2}-8 x-7
  1. Expand the given function: We need to expand the given function f(x)=−(x+1)(x+7)f(x) = -(x+1)(x+7) to find the equivalent form that shows the y-intercept.\newlinef(x)=−(x2+7x+x+7)f(x) = -(x^2 + 7x + x + 7)\newlinef(x)=−(x2+8x+7)f(x) = -(x^2 + 8x + 7)\newlineNow, distribute the negative sign.\newlinef(x)=−x2−8x−7f(x) = -x^2 - 8x - 7
  2. Find the y-intercept: The y-intercept of a function is the value of f(x)f(x) when x=0x = 0. Let's find the y-intercept of the function f(x)=−x2−8x−7f(x) = -x^2 - 8x - 7.\newlinef(0)=−(0)2−8(0)−7f(0) = -(0)^2 - 8(0) - 7\newlinef(0)=−7f(0) = -7\newlineThe y-intercept is −7-7.
  3. Compare with given choices: Now, we need to compare the expanded form of f(x)f(x) with the given choices to see which one has −7-7 as the constant or coefficient that represents the y-intercept.\newline(A) f(x)=−(x+4)2+9f(x) = -(x+4)^2 + 9 does not have −7-7 as a constant or coefficient.\newline(B) f(x)=−(x+4)2−9f(x) = -(x+4)^2 - 9 does not have −7-7 as a constant or coefficient.\newline(C) f(x)=−x2+8x+7f(x) = -x^2 + 8x + 7 does not have −7-7 as a constant or coefficient.\newline(D) f(x)=−x2−8x−7f(x) = -x^2 - 8x - 7 has −7-7 as a constant, which is the y-intercept we found.
  4. Identify the correct answer: The correct answer is (D) f(x)=−x2−8x−7f(x) = -x^2 - 8x - 7 because it is the equivalent form of the given function and it shows the yy-intercept as a constant or coefficient.

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