Question

write the equation in vertex form for the parabola with vertex (-2, 0) and focus (6, 0). simplify any fractions.

Explanation:

Step1: Find the value of \(p\)

The formula for the vertex form of a parabola that opens left - right is \((y - k)^2=4p(x - h)\), where \((h,k)\) is the vertex. The distance between the vertex \((- 2,0)\) and the focus \((6,0)\) is \(p\). Using the distance formula for points on the x - axis (since \(y\) - coordinates are the same), \(p=x_{focus}-x_{vertex}\). So, \(p = 6-(-2)=8\).

Step2: Write the vertex form of the parabola

The vertex of the parabola is \((h,k)=(-2,0)\). Substitute \(h=-2\), \(k = 0\), and \(p = 8\) into the equation \((y - k)^2=4p(x - h)\). We get \((y-0)^2=4\times8(x + 2)\).

Answer:

\(y^{2}=32(x + 2)\)