Question

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 the focus \((0,8)\) has the same \(x -\)coordinate as the vertex, the parabola is of the form \((x - h)^2=4p(y - k)\), where \((h,k)\) is the vertex.

Step2: Calculate the value of \(p\)

The distance between the vertex \((0,7)\) and the focus \((0,8)\) is \(p\). Using the distance formula \(p=\vert y_{focus}-y_{vertex}\vert\), we have \(p=\vert8 - 7\vert=1\)

Step3: Substitute \(h\), \(k\), and \(p\) into the formula

Substitute \(h = 0\), \(k = 7\), and \(p = 1\) into \((x - h)^2=4p(y - k)\)

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Answer:

\(x^{2}=4(y - 7)\)