Question

the graph to the right shows a parabola along with its vertex, focus, and directrix. determine the coordinates of the vertex and focus, and the equation of the directrix. what are the coordinates of the vertex? (-4, -1) (type an ordered pair.) what are the coordinates of the focus? (type an ordered pair.)

Explanation:

Step1: Analyze the distance between vertex and directrix

The vertex is \((-4,-1)\). The directrix is \(y = - 0.5\). The distance \(p\) between the vertex \((h,k)=(-4,-1)\) and the directrix \(y=k + p\) (for a parabola of the form \((x - h)^{2}=4p(y - k)\)) is calculated as \(p=-0.5-(-1)= - 0.5\)

Step2: Find the focus coordinates

For a parabola \((x - h)^{2}=4p(y - k)\), the focus is at \((h,k + p)\). Substituting \(h=-4,k=-1,p=-0.5\), we get \(k + p=-1+(-0.5)=-1.5\)

Answer:

\((-4,-1.5)\)