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Write the equation in vertex form for the parabola with vertex (0,−6)(0,-6) and directrix y=−2y = -2.\newlineSimplify any fractions.\newline______

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Q. Write the equation in vertex form for the parabola with vertex (0,−6)(0,-6) and directrix y=−2y = -2.\newlineSimplify any fractions.\newline______
  1. Vertex and Directrix Information: Vertex: (0,−6)(0,-6)\newlineDirectrix: y=−2y = -2\newlineSince the directrix is horizontal, the parabola is vertical.
  2. Vertex Form of Parabola: Vertex form of a vertical parabola: y=a(x−h)2+ky = a(x-h)^2 + k
  3. Direction of Parabola: The vertex is below the directrix, so the parabola opens upwards.
  4. Calculate Distance: Distance between vertex and directrix: ∣−6−(−2)∣=4|-6 - (-2)| = 4
  5. Calculate Value of aa: Value of aa is 14×distance\frac{1}{4 \times \text{distance}} since the parabola opens upwards.\newlinea=14×4=116a = \frac{1}{4 \times 4} = \frac{1}{16}
  6. Substitute Values into Vertex Form: Substitute a=116a = \frac{1}{16}, h=0h = 0, and k=−6k = -6 into the vertex form.\newliney=116(x−0)2−6y = \frac{1}{16}(x - 0)^2 - 6\newliney=116x2−6y = \frac{1}{16}x^2 - 6

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