Question

write the equation in vertex form for the parabola with vertex (0,1) and focus (0,4). simplify any fractions.

Explanation:

Step1: Determine the direction and value of \(p\)

Since the vertex \((h,k)=(0,1)\) and focus \((h,k + p)=(0,4)\), we have \(k + p=4\). Substituting \(k = 1\), we get \(1 + p=4\), so \(p=3\). The parabola opens upwards.

Step2: Use the vertex - form formula

The vertex - form of a parabola that opens up or down is \(y=a(x - h)^{2}+k\), and for a parabola \(a=\frac{1}{4p}\). Substituting \(h = 0,k = 1,p = 3\) into the formula, we have \(a=\frac{1}{4\times3}=\frac{1}{12}\).

Step3: Write the equation

Substitute \(h = 0,k = 1,a=\frac{1}{12}\) into \(y=a(x - h)^{2}+k\). The equation is \(y=\frac{1}{12}(x - 0)^{2}+1\), which simplifies to \(y=\frac{1}{12}x^{2}+1\).

Answer:

\(y=\frac{1}{12}x^{2}+1\)