Question

write the equation in vertex form for the parabola with vertex (0, -6) and focus (0, -4). simplify any fractions.

Explanation:

Step1: Recall the vertex form of a parabola

The vertex form of a parabola with vertex \((h,k)\) is \(x^{2}=4p(y - k)\) (for parabolas opening up or down). Here, the vertex is \((h,k)=(0,-6)\).

Step2: Calculate the value of \(p\)

The formula for \(p\) is \(p=y_{\text{focus}}-y_{\text{vertex}}\). Given \(y_{\text{focus}}=-4\) and \(y_{\text{vertex}}=-6\), then \(p=-4-(-6)=2\).

Step3: Substitute \(h\), \(k\), and \(p\) into the vertex - form equation

Substitute \(h = 0\), \(k=-6\), and \(p = 2\) into \(x^{2}=4p(y - k)\). We get \(x^{2}=4\times2(y+6)\).

Answer:

\(x^{2}=8(y + 6)\)