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Write the equation in vertex form for the parabola with vertex (0,4)(0,4) and focus (0,2)(0,2).\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 (0,2)(0,2).\newlineSimplify any fractions.\newline______
  1. Identify Orientation: Identify the orientation of the parabola based on the vertex and focus.\newlineSince the vertex and focus have the same xx-coordinate, the parabola is vertical.
  2. Determine Opening Direction: Determine the direction the parabola opens.\newlineThe focus is at (0,2)(0,2), which is below the vertex (0,4)(0,4), so the parabola opens downward.
  3. Calculate Value of 'a': Calculate the value of 'a' using the distance between the vertex and focus.\newlineDistance = ∣4−2∣=2|4 - 2| = 2\newlineSince the parabola opens downward, 'a' is negative.\newlinea=−14×distance=−14×2=−18a = -\frac{1}{4 \times \text{distance}} = -\frac{1}{4 \times 2} = -\frac{1}{8}
  4. Write Equation in Vertex Form: Write the equation in vertex form using the vertex (h,k)=(0,4)(h,k) = (0,4) and the value of ′a′'a'.y=a(x−h)2+ky = a(x-h)^2 + ky=−18(x−0)2+4y = -\frac{1}{8}(x-0)^2 + 4y=−18x2+4y = -\frac{1}{8}x^2 + 4

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