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A parabola opening up or down has vertex (0,−7)(0,-7) and passes through (8,1)(8,1). Write its equation in vertex form.\newlineSimplify any fractions.

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Full solution

Q. A parabola opening up or down has vertex (0,−7)(0,-7) and passes through (8,1)(8,1). Write its equation in vertex form.\newlineSimplify any fractions.
  1. Identify Vertex Values: We have:\newlineVertex: (0,−7)(0,-7)\newlineIdentify the values of hh and kk.\newlineVertex is (0,−7)(0,-7).\newlineh=0h = 0\newlinek=−7k = -7
  2. Select Equation: We have:\newliney=a(x−h)2+ky = a(x - h)^2 + k\newlineh=0h = 0 and k=−7k = -7\newlineSelect the equation after substituting the values of hh and kk.\newlineSubstitute h=0h = 0 and k=−7k = -7 in y=a(x−h)2+ky = a(x - h)^2 + k.\newliney=a(x−0)2−7y = a(x - 0)^2 - 7\newliney=ax2−7y = ax^2 - 7
  3. Find Value of a: We have: y=ax2−7y = ax^2 - 7
    Point: (8,1)(8,1)
    Find the value of a.
    y=ax2−7y = ax^2 - 7
    1=a(8)2−71 = a(8)^2 - 7
    1=64a−71 = 64a - 7
    1+7=64a1 + 7 = 64a
    8=64a8 = 64a
    864=a\frac{8}{64} = a
    a=18a = \frac{1}{8}
  4. Write Vertex Form: We found:\newlinea=18a = \frac{1}{8}\newlineh=0h = 0\newlinek=−7k = -7\newlineWrite the equation of a parabola in vertex form.\newliney=a(x−h)2+ky = a(x - h)^2 + k\newliney=18(x−0)2−7y = \frac{1}{8}(x - 0)^2 - 7\newlineVertex form: y=18x2−7y = \frac{1}{8}x^2 - 7

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