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A parabola opening up or down has vertex (0,1)(0,1) and passes through (−8,9)(-8,9). Write its equation in vertex form.\newlineSimplify any fractions.

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Q. A parabola opening up or down has vertex (0,1)(0,1) and passes through (−8,9)(-8,9). 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 (0,1)(0, 1)?\newlineSubstitute 00 for hh and 11 for kk in the vertex form.\newliney=a(x−0)2+1y = a(x - 0)^2 + 1\newliney=ax2+1y = ax^2 + 1
  3. Use Point to Find 'a': Use the point (−8,9)(-8, 9) to find the value of 'a'.\newlineReplace the variables with (−8,9)(-8, 9) in the equation.\newlineSubstitute −8-8 for xx and 99 for yy.\newline9=a(−8)2+19 = a(-8)^2 + 1\newline9=64a+19 = 64a + 1
  4. Solve for 'a': Solve for 'a'.\newline9=64a+19 = 64a + 1\newlineSubtract 11 from both sides.\newline8=64a8 = 64a\newlineDivide both sides by 6464.\newline864=a\frac{8}{64} = a\newline18=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+1y = ax^2 + 1.\newliney=(18)x2+1y = \left(\frac{1}{8}\right)x^2 + 1\newlineVertex form of the parabola: y=(18)x2+1y = \left(\frac{1}{8}\right)x^2 + 1

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