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Write the equation in vertex form for the parabola with vertex (0,5)(0,5) and focus (0,7)(0,7).\newlineSimplify any fractions.\newline______

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Q. Write the equation in vertex form for the parabola with vertex (0,5)(0,5) and focus (0,7)(0,7).\newlineSimplify any fractions.\newline______
  1. Identify vertex form: Identify the vertex form of a vertical parabola.\newlineVertex form: y=a(x−h)2+ky = a(x-h)^2 + k
  2. Given vertex and focus: Given vertex (h,k)=(0,5)(h,k) = (0,5) and focus (0,7)(0,7).\newlineSince the focus is above the vertex, the parabola opens upwards.
  3. Calculate distance for 'a': Calculate the distance between the vertex and the focus to find the value of 'a'.\newlineDistance = ∣7−5∣=2|7 - 5| = 2\newlineThe distance is the same as 14a\frac{1}{4a}, so 2=14a2 = \frac{1}{4a}.
  4. Solve for 'a': Solve for 'a'.\newline14a=2\frac{1}{4a} = 2\newlinea=18a = \frac{1}{8}
  5. Substitute into equation: Substitute aa, hh, and kk into the vertex form equation.y=(18)(x−0)2+5y = \left(\frac{1}{8}\right)(x-0)^2 + 5y=(18)x2+5y = \left(\frac{1}{8}\right)x^2 + 5

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