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Write the equation in vertex form for the parabola with vertex (0,2)(0,2) and directrix y=5y = 5.\newlineSimplify any fractions.\newline______

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Q. Write the equation in vertex form for the parabola with vertex (0,2)(0,2) and directrix y=5y = 5.\newlineSimplify any fractions.\newline______
  1. Identify Orientation: Identify the orientation of the parabola.\newlineSince the directrix is horizontal y=5y=5, the parabola is vertical.
  2. Determine Direction: Determine the direction the parabola opens.\newlineThe vertex (0,2)(0,2) is below the directrix y=5y=5, so the parabola opens downward.
  3. Calculate Distance: Calculate the distance between the vertex and the directrix.\newlineDistance = ∣2−5∣=3|2 - 5| = 3.
  4. Find Value of a: Find the value of aa using the distance.\newlineThe distance is equal to 14a\frac{1}{4a} for a vertical parabola that opens downward, so a=−14×3a = -\frac{1}{4\times 3}.
  5. Simplify Value of a: Simplify the value of a.\newlinea=−112a = -\frac{1}{12}.
  6. Write Equation: Write the equation in vertex form.\newlineVertex form is y=a(x−h)2+ky = a(x-h)^2 + k.\newlineSubstitute a=−112a = -\frac{1}{12}, h=0h = 0, and k=2k = 2.\newliney=−112(x−0)2+2y = -\frac{1}{12}(x-0)^2 + 2.
  7. Simplify Equation: Simplify the equation. y=−112x2+2y = -\frac{1}{12}x^2 + 2.

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