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The equation of a parabola is y=x2+2x−1y = x^2 + 2x - 1. Write the equation in vertex form.\newlineWrite any numbers as integers or simplified proper or improper fractions.\newline______

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Q. The equation of a parabola is y=x2+2x−1y = x^2 + 2x - 1. Write the equation in vertex form.\newlineWrite any numbers as integers or simplified proper or improper fractions.\newline______
  1. Identify Vertex Form: Identify the vertex form of a parabola. The vertex 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.
  2. Complete Square: Complete the square to rewrite the equation y=x2+2x−1y = x^2 + 2x - 1 in vertex form.\newlineFirst, we need to complete the square for the xx-terms. To do this, we take half of the coefficient of xx, which is 22, divide it by 22 to get 11, and then square it to get 11. We will add and subtract this value inside the equation.
  3. Add/Subtract Square: Add and subtract the square of half the coefficient of xx inside the equation.\newlineThe equation becomes y=x2+2x+1−1−1y = x^2 + 2x + 1 - 1 - 1. We added 11 and subtracted 11 to complete the square, and we subtracted an additional 11 because it was already present in the original equation.
  4. Group and Combine: Group the perfect square trinomial and combine the constants.\newlineThe equation now is y=(x2+2x+1)−2y = (x^2 + 2x + 1) - 2. The expression (x2+2x+1)(x^2 + 2x + 1) is a perfect square trinomial and can be written as (x+1)2(x + 1)^2. The constants −1-1 and −1-1 combine to give −2-2.
  5. Write in Vertex Form: Write the equation in vertex form.\newlineThe equation in vertex form is y=(x+1)2−2y = (x + 1)^2 - 2. This is the vertex form of the given parabola, where the vertex (h,k)(h, k) is (−1,−2)(-1, -2).

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