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The graph of y=f(x) is a parabola in the xy-plane that is symmetric with respect to the line x=-2. The y-coordinate of the vertex of the graph of f is a maximum function value. Which of the following equations could represent function f ? Choose 1 answer: (A) f(x)=5(x-2)^(2)+3 (B) f(x)=5(x+2)^(2)+3 (C) f(x)=-5(x-2)^(2)+3 (D) f(x)=-5(x+2)^(2)+3

The graph of y=f(x) y=f(x) is a parabola in the xy x y -plane that is symmetric with respect to the line x=−2 x=-2 . The y y -coordinate of the vertex of the graph of f f is a maximum function value. Which of the following equations could represent function f f ?\newlineChoose 11 answer:\newline(A) f(x)=5(x−2)2+3 f(x)=5(x-2)^{2}+3 \newline(B) f(x)=5(x+2)2+3 f(x)=5(x+2)^{2}+3 \newline(C) f(x)=−5(x−2)2+3 f(x)=-5(x-2)^{2}+3 \newline(D) f(x)=−5(x+2)2+3 f(x)=-5(x+2)^{2}+3

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Q. The graph of y=f(x) y=f(x) is a parabola in the xy x y -plane that is symmetric with respect to the line x=−2 x=-2 . The y y -coordinate of the vertex of the graph of f f is a maximum function value. Which of the following equations could represent function f f ?\newlineChoose 11 answer:\newline(A) f(x)=5(x−2)2+3 f(x)=5(x-2)^{2}+3 \newline(B) f(x)=5(x+2)2+3 f(x)=5(x+2)^{2}+3 \newline(C) f(x)=−5(x−2)2+3 f(x)=-5(x-2)^{2}+3 \newline(D) f(x)=−5(x+2)2+3 f(x)=-5(x+2)^{2}+3
  1. Identify Parabola Form: Identify the general form of a parabola with its vertex at the point (h,k)(h, k). The general form of a parabola is given by y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. If the parabola opens upwards, aa is positive, and if it opens downwards, aa is negative. Since the y-coordinate of the vertex is a maximum, the parabola opens downwards, which means aa should be negative.
  2. Determine Symmetry Line: Determine the value of hh from the symmetry line.\newlineThe line of symmetry of the parabola is given by x=hx = h. Since the parabola is symmetric with respect to the line x=−2x = -2, the value of hh is −2-2.
  3. Eliminate Incorrect Options: Eliminate the options that do not have the correct vertex form with h=−2h = -2.\newlineOption (A) f(x)=5(x−2)2+3f(x) = 5(x - 2)^2 + 3 has h=2h = 2, which does not match the required line of symmetry x=−2x = -2.\newlineOption (B) f(x)=5(x+2)2+3f(x) = 5(x + 2)^2 + 3 has h=−2h = -2, which matches the required line of symmetry.\newlineOption (C) f(x)=−5(x−2)2+3f(x) = -5(x - 2)^2 + 3 has h=2h = 2, which does not match the required line of symmetry x=−2x = -2.\newlineOption (D) f(x)=−5(x+2)2+3f(x) = -5(x + 2)^2 + 3 has h=−2h = -2, which matches the required line of symmetry.
  4. Identify Parabola Direction: Identify the correct option based on the direction of the parabola.\newlineSince the yy-coordinate of the vertex is a maximum, the parabola must open downwards, which means aa should be negative. This eliminates option (B) because it has a positive aa value.
  5. Choose Correct Answer: Choose the correct answer from the remaining options.\newlineOption (D) f(x)=−5(x+2)2+3f(x) = -5(x + 2)^2 + 3 is the only remaining option that satisfies both conditions: it has the correct line of symmetry (h=−2h = -2) and the correct direction of the parabola (aa is negative).

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