Question

write the equation in vertex form for the parabola with vertex (2, 0) and directrix x = 8. simplify any fractions.

Explanation:

Step1: Recall the formula for a parabola

The vertex form of a parabola with a horizontal directrix is \((y - k)^2 = 4p(x - h)\), where \((h,k)\) is the vertex. The distance from the vertex to the directrix is \(|p|\).

Step2: Determine the value of \(p\)

The vertex is \((h,k)=(2,0)\) and the directrix is \(x = 8\). Since the directrix \(x=8\) and the vertex \(x\) - coordinate is \(h = 2\), and for a parabola \(x=h - p\) (because the directrix is to the right of the vertex, \(p<0\)). So \(8=2 - p\), then \(p=2 - 8=-6\).

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

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

Answer:

\(y^{2}=-24(x - 2)\)