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A parabola opening up or down has vertex (0,−2)(0,-2) and passes through (12,−20)(12,-20). Write its equation in vertex form.\newlineSimplify any fractions.

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Q. A parabola opening up or down has vertex (0,−2)(0,-2) and passes through (12,−20)(12,-20). Write its equation in vertex form.\newlineSimplify any fractions.
  1. Vertex Form Explanation: What is the vertex form of the 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. Equation with Vertex: What is the equation of a parabola with a vertex at (0,−2)(0, -2)?\newlineSince the vertex is (0,−2)(0, -2), we substitute h=0h = 0 and k=−2k = -2 into the vertex form equation.\newliney=a(x−0)2−2y = a(x - 0)^2 - 2\newliney=ax2−2y = ax^2 - 2
  3. Value of 'a' Calculation: Determine the value of 'a' using the point (12,−20)(12, -20).\newlineWe know the parabola passes through the point (12,−20)(12, -20), so we substitute x=12x = 12 and y=−20y = -20 into the equation to find 'a'.\newline−20=a(12)2−2-20 = a(12)^2 - 2
  4. Solving for 'a': Solve for 'a'.\newline−20=144a−2-20 = 144a - 2\newlineAdd 22 to both sides to isolate the term with 'a'.\newline−18=144a-18 = 144a\newlineDivide both sides by 144144 to solve for 'a'.\newlinea=−18144a = -\frac{18}{144}\newlinea=−18a = -\frac{1}{8}
  5. Final Equation in Vertex Form: Write the equation of the parabola in vertex form using the value of aa. Now that we have found a=−18a = -\frac{1}{8}, we substitute it back into the equation y=ax2−2y = ax^2 - 2. y=(−18)x2−2y = (-\frac{1}{8})x^2 - 2 This is the equation of the parabola in vertex form.

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