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Write the equation in vertex form for the parabola with vertex (0,−7)(0,-7) and directrix y=−4y = -4.\newlineSimplify any fractions.\newline______

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Q. Write the equation in vertex form for the parabola with vertex (0,−7)(0,-7) and directrix y=−4y = -4.\newlineSimplify any fractions.\newline______
  1. Identify orientation: Identify the orientation of the parabola.\newlineSince the directrix is horizontal y=−4y = -4, the parabola is vertical.
  2. Determine opening direction: Determine the direction the parabola opens.\newlineThe vertex (0,−7)(0,-7) is below the directrix y=−4y = -4, so the parabola opens upward.
  3. Calculate distance to directrix: Calculate the distance between the vertex and the directrix.\newlineDistance = ∣−7−(−4)∣=∣−7+4∣=∣−3∣=3| -7 - (-4) | = |-7 + 4| = | -3 | = 3
  4. Find value of aa: Find the value of aa using the distance.\newlineThe distance is equal to 14a\frac{1}{4a}, so 3=14a3 = \frac{1}{4a}.\newlineSolve for aa: a=14×3=112a = \frac{1}{4\times 3} = \frac{1}{12}.
  5. Write equation in vertex form: Write the equation in vertex form.\newlineVertex form for a vertical parabola is y=a(x−h)2+ky = a(x-h)^2 + k.\newlineSubstitute a=112a = \frac{1}{12}, h=0h = 0, and k=−7k = -7.\newliney=112(x−0)2−7y = \frac{1}{12}(x - 0)^2 - 7\newliney=112x2−7y = \frac{1}{12}x^2 - 7

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