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The equation of a parabola is y=x2−2x+2y = x^2 - 2x + 2. 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+2y = x^2 - 2x + 2. 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 the Square: Complete the square to rewrite the given equation y=x2−2x+2y = x^2 - 2x + 2 in vertex form.\newlineFirst, we need to create a perfect square trinomial from the quadratic part of the equation. To do this, we take half of the coefficient of xx, which is −2-2, divide it by 22 to get −1-1, and then square it to get 11. We will add and subtract this value inside the equation.
  3. Add and Subtract: Add and subtract the square of half the coefficient of xx inside the equation.\newlineThe equation becomes y=x2−2x+1−1+2y = x^2 - 2x + 1 - 1 + 2.\newlineWe added and subtracted 11 to complete the square while keeping the equation balanced.
  4. Rewrite Equation: Rewrite the equation by grouping the perfect square trinomial and combining the constants.\newlineThe equation now looks like y=(x2−2x+1)+(2−1)y = (x^2 - 2x + 1) + (2 - 1).\newlineThis simplifies to y=(x−1)2+1y = (x - 1)^2 + 1, which is now in vertex form.

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