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 vertex form of a parabola

The vertex form of a parabola with vertex \((h,k)\) is \((y - k)^2=4p(x - h)\). Here, the vertex is \((h,k)=(2,0)\).

Step2: Find the value of \(p\)

The distance between the vertex \((2,0)\) and the directrix \(x = 8\) is \(|h-p|=|2 - p|=8 - 2\) (since for a parabola of the form \((y - k)^2=4p(x - h)\), the directrix is \(x=h - p\)). Solving \(|2 - p|=6\), we get \(p=- 6\) (because the parabola opens to the left as the directrix \(x = 8\) is to the right of the vertex \((2,0)\)).

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

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:

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