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Write the equation in vertex form for the parabola with vertex (0,−2)(0,-2) and focus (0,−8)(0,-8).\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 focus (0,−8)(0,-8).\newlineSimplify any fractions.\newline______
  1. Identify Parabola Orientation: Vertex: (0,−2)(0,-2)\newlineFocus: (0,−8)(0,-8)\newlineIdentify whether the parabola is vertical or horizontal.\newlineSince the xx-coordinates of the vertex and focus are the same, the parabola is vertical.
  2. Vertex Form Explanation: Vertex form of a vertical parabola: y=a(x−h)2+ky = a(x-h)^2+k Here, (h,k)(h,k) is the vertex.
  3. Determine Parabola Direction: Vertex: (0,−2)(0,-2)\newlineFocus: (0,−8)(0,-8)\newlineDetermine if the parabola opens upward or downward.\newlineSince the focus is below the vertex, the parabola opens downward.
  4. Calculate Distance for 'a': Vertex: (0,−2)(0,-2)\newlineFocus: (0,−8)(0,-8)\newlineCalculate the distance between vertex and focus to find the value of 'aa'.\newlineDistance: ∣−8−(−2)∣=∣−8+2∣=6|-8 - (-2)| = |-8 + 2| = 6
  5. Calculate 'a' Value: Distance between the vertex and focus: 66\newlineThe value of 'a' is negative because the parabola opens downward.\newlinea=−14pa = -\frac{1}{4p}, where pp is the distance from the vertex to the focus.\newlinea=−14×6a = -\frac{1}{4\times 6}\newlinea=−124a = -\frac{1}{24}
  6. Substitute Values in Equation: We found:\newlinea=−124a = -\frac{1}{24}\newlineVertex (h,k)=(0,−2)(h,k) = (0,-2)\newlineSubstitute −124-\frac{1}{24} for aa, 00 for hh and −2-2 for kk in the vertex form equation.\newliney=−124(x−0)2−2y = -\frac{1}{24}(x-0)^2-2\newliney=−124x2−2y = -\frac{1}{24} x^2 - 2

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