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The equation of a parabola is y=x2−8x+7y = x^2 - 8x + 7. 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=x2−8x+7y = x^2 - 8x + 7. 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 Transformation: Complete the square to transform the given equation into vertex form.\newlineWe start with the given equation y=x2−8x+7y = x^2 - 8x + 7. To complete the square, we need to find the value that makes x2−8xx^2 - 8x a perfect square trinomial. We do this by taking half of the coefficient of xx, squaring it, and adding it to and subtracting it from the equation.\newlineHalf of −8-8 is −4-4, and (−4)2=16(-4)^2 = 16. So we add and subtract 1616 to the equation.\newliney=x2−8x+16−16+7y = x^2 - 8x + 16 - 16 + 7
  3. Rewrite and Simplify: Rewrite the equation with the perfect square trinomial and simplify.\newlineNow we have y=(x2−8x+16)−16+7y = (x^2 - 8x + 16) - 16 + 7. The expression in the parentheses is a perfect square trinomial, which can be written as (x−4)2(x - 4)^2.\newlineSo the equation becomes y=(x−4)2−16+7y = (x - 4)^2 - 16 + 7.\newlineSimplifying the constants gives us y=(x−4)2−9y = (x - 4)^2 - 9.

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