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Write the equation in vertex form for the parabola with vertex (0,4)(0,4) and focus (0,0)(0,0).\newlineSimplify any fractions.\newline______

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Q. Write the equation in vertex form for the parabola with vertex (0,4)(0,4) and focus (0,0)(0,0).\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)=(0,4)(h,k) = (0,4) and focus at (0,0)(0,0). Since the focus is below the vertex, the parabola opens downward.
  3. Calculate distance and 'a': Calculate the distance between vertex and focus to find the value of 'a'.\newlineDistance = ∣4−0∣=4|4 - 0| = 4\newlineSince the parabola opens downward, 'a' is negative.\newlinea=−14pa = -\frac{1}{4p}, where pp is the distance from the vertex to the focus.\newlinea=−14×4a = -\frac{1}{4\times 4}\newlinea=−116a = -\frac{1}{16}
  4. Substitute values into equation: Substitute aa value, and vertex coordinates into the vertex form equation.y=a(x−h)2+ky = a(x-h)^2+ky=−116(x−0)2+4y = -\frac{1}{16}(x-0)^2+4y=−116x2+4y = -\frac{1}{16}x^2+4

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