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Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an (x,y) point. y=-2x^(2)+8x Answer:

Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an (x,y) (x, y) point.\newliney=−2x2+8x y=-2 x^{2}+8 x \newlineAnswer:

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Q. Find the coordinates of the vertex of the following parabola algebraically. Write your answer as an (x,y) (x, y) point.\newliney=−2x2+8x y=-2 x^{2}+8 x \newlineAnswer:
  1. Factor out x terms: 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. To find the vertex of the parabola given by y=−2x2+8xy = -2x^2 + 8x, we need to complete the square to convert the equation into vertex form.
  2. Complete the square: First, factor out the coefficient of x2x^2 from the xx terms: y=−2(x2−4x)y = -2(x^2 - 4x). Leave a space to add and subtract the necessary term to complete the square.
  3. Simplify the equation: To complete the square, take half of the coefficient of xx, which is 44, square it, and add and subtract it inside the parentheses: y=−2(x2−4x+(4/2)2−(4/2)2)y = -2(x^2 - 4x + (4/2)^2 - (4/2)^2).
  4. Factor the trinomial: Simplify the equation by adding the square and its negative inside the parentheses: y=−2(x2−4x+4−4)y = -2(x^2 - 4x + 4 - 4).
  5. Distribute −2-2: Now, factor the perfect square trinomial inside the parentheses and simplify the constant term: y=−2((x−2)2−4)y = -2((x - 2)^2 - 4).
  6. Final vertex form: Distribute the −2-2 to the terms inside the parentheses: y=−2(x−2)2+8y = -2(x - 2)^2 + 8.
  7. Final vertex form: Distribute the −2-2 to the terms inside the parentheses: y=−2(x−2)2+8y = -2(x - 2)^2 + 8.Now we have the equation in vertex form: y=−2(x−2)2+8y = -2(x - 2)^2 + 8. The vertex (h,k)(h, k) can be read directly from the equation as (2,8)(2, 8).

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