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Write the equation in vertex form for the parabola with vertex (0,−2)(0,-2) and directrix y=6y = 6.\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=6y = 6.\newlineSimplify any fractions.\newline______
  1. Identify Parabola Orientation: Since the directrix is y=6y = 6, the parabola is vertical and opens downward because the vertex (0,−2)(0,-2) is below the directrix.
  2. Vertex Form of Parabola: The vertex form of a vertical parabola is y=a(x−h)2+ky = a(x-h)^2 + k, where (h,k)(h,k) is the vertex.
  3. Calculate Distance to Directrix: The distance between the vertex and the directrix is ∣6−(−2)∣=8|6 - (-2)| = 8.
  4. Determine Focus Value: Since the parabola opens downward, aa is negative. The focus is the same distance from the vertex as the directrix, so a=−14⋅8=−132a = -\frac{1}{4\cdot8} = -\frac{1}{32}.
  5. Substitute Values into Equation: Substitute a=−132a = -\frac{1}{32} and the vertex (h,k)=(0,−2)(h,k) = (0,-2) into the vertex form equation: y=−132(x−0)2−2y = -\frac{1}{32}(x-0)^2 - 2.
  6. Simplify Equation: Simplify the equation: y=−132x2−2y = -\frac{1}{32} x^2 - 2.

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