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Write the equation in vertex form for the parabola with vertex (0,−6)(0,-6) and directrix y=−7y = -7.\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=−7y = -7.\newlineSimplify any fractions.\newline______
  1. Identify Orientation: Identify the orientation of the parabola.\newlineSince the directrix is horizontal y=−7y = -7, the parabola is vertical.
  2. Determine Opening Direction: Determine the direction the parabola opens. The vertex (0,−6)(0,-6) is above the directrix y=−7y = -7, so the parabola opens upward.
  3. Find Distance to Directrix: Find the distance between the vertex and the directrix.\newlineDistance = ∣−7−(−6)∣=∣−7+6∣=∣1∣=1|-7 - (-6)| = |-7 + 6| = |1| = 1.
  4. Calculate Value of a: Calculate the value of a using the distance.\newlineThe distance is equal to 1/(4a)1/(4a), so 1=1/(4a)1 = 1/(4a).\newlineSolving for a gives a=1/4a = 1/4.
  5. Write Equation in Vertex Form: Write the equation in vertex form.\newlineThe vertex form is y=a(x−h)2+ky = a(x-h)^2 + k.\newlineSubstitute a=14a = \frac{1}{4}, h=0h = 0, and k=−6k = -6 into the equation.\newliney=14(x−0)2−6y = \frac{1}{4}(x-0)^2 - 6.
  6. Simplify Equation: Simplify the equation. y=14x2−6y = \frac{1}{4}x^2 - 6.

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