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A parabola opening up or down has vertex (0,6)(0,6) and passes through (8,−2)(8,-2). Write its equation in vertex form.\newlineSimplify any fractions.

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Q. A parabola opening up or down has vertex (0,6)(0,6) and passes through (8,−2)(8,-2). Write its equation in vertex form.\newlineSimplify any fractions.
  1. Vertex Form of Parabola: 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 (0,6)(0, 6)?\newlineSubstitute 00 for hh and 66 for kk in the vertex form.\newliney=a(x−0)2+6y = a(x - 0)^2 + 6\newliney=ax2+6y = ax^2 + 6
  3. Find 'a' Using Point: Use the point (8,−2)(8, -2) to find the value of 'a'.\newlineReplace the variables with (8,−2)(8, -2) in the equation.\newlineSubstitute 88 for xx and −2-2 for yy.\newline−2=a(8)2+6-2 = a(8)^2 + 6\newline−2=64a+6-2 = 64a + 6
  4. Solve for 'a': Solve for 'a'.\newline−2=64a+6-2 = 64a + 6\newlineSubtract 66 from both sides.\newline−2−6=64a-2 - 6 = 64a\newline−8=64a-8 = 64a\newlineDivide both sides by 6464.\newline−864=a-\frac{8}{64} = a\newlineSimplify the fraction.\newline−18=a-\frac{1}{8} = a
  5. Write Equation with 'a': Write the equation of the parabola with the value of 'a'.\newlineSubstitute −18-\frac{1}{8} for aa in the equation y=ax2+6y = ax^2 + 6.\newliney=(−18)x2+6y = \left(-\frac{1}{8}\right)x^2 + 6\newlineVertex form of the parabola: y=−(18)x2+6y = -\left(\frac{1}{8}\right)x^2 + 6

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