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A parabola opening up or down has vertex (0,−3)(0,-3) and passes through (8,−19)(8,-19). Write its equation in vertex form.\newlineSimplify any fractions.

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Q. A parabola opening up or down has vertex (0,−3)(0,-3) and passes through (8,−19)(8,-19). Write its equation in vertex form.\newlineSimplify any fractions.
  1. Identify vertex form: Identify the vertex form of a parabola.\newlineThe vertex form of a parabola is given by the equation y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  2. Plug vertex coordinates: Plug the vertex coordinates into the vertex form.\newlineSince the vertex is given as (0,−3)(0, -3), we substitute h=0h = 0 and k=−3k = -3 into the vertex form equation.\newliney=a(x−0)2−3y = a(x - 0)^2 - 3\newliney=ax2−3y = ax^2 - 3
  3. Use point to find 'a': Use the point (8,−19)(8, -19) to find the value of 'a'.\newlineWe know the parabola passes through the point (8,−19)(8, -19), so we can substitute x=8x = 8 and y=−19y = -19 into the equation to solve for 'a'.\newline−19=a(8)2−3-19 = a(8)^2 - 3\newline−19=64a−3-19 = 64a - 3
  4. Solve for 'a': Solve for 'a'.\newlineNow we isolate 'a' by adding 33 to both sides and then dividing by 6464.\newline−19+3=64a-19 + 3 = 64a\newline−16=64a-16 = 64a\newlinea=−16/64a = -16 / 64\newlinea=−1/4a = -1 / 4
  5. Write final equation: Write the final equation of the parabola in vertex form.\newlineNow that we have the value of aa, we can write the equation of the parabola:\newliney=(−14)(x−0)2−3y = (-\frac{1}{4})(x - 0)^2 - 3\newlineSimplifying further, we get:\newliney=−14x2−3y = -\frac{1}{4}x^2 - 3

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