write the equation in vertex form for the parabola with vertex (0,7) and focus (0,8). simplify any fractions.

Explanation:

Step1: Determine the form of the parabola

Since the vertex ((h,k)=(0,7)) and focus ((0,8)) have the same (x -)coordinate, the parabola is vertical. The vertex - form of a vertical parabola is (y=a(x - h)^{2}+k).

Step2: Calculate the value of (a)

The distance between the vertex ((h,k)) and the focus ((h,k + p)) is (p). Here, (h = 0,k = 7), and the focus is ((0,8)). So, (k+p=8), substituting (k = 7) gives (p=1). For a parabola (y=a(x - h)^{2}+k), (a=\frac{1}{4p}). Substituting (p = 1) into (a=\frac{1}{4p}), we get (a=\frac{1}{4}).

Step3: Write the equation of the parabola

Substitute (h = 0,k = 7,a=\frac{1}{4}) into the vertex - form (y=a(x - h)^{2}+k). The equation is (y=\frac{1}{4}(x - 0)^{2}+7).

Answer:

(y=\frac{1}{4}x^{2}+7)