Question

determine the vertex, focus, directrix, and endpoints of the latus rectum for the parabola (y^{2}=-8x). enter your answers as points ((a,b)). enter the end points of the latus rectum in increasing order of the (y -)coordinate. the vertex is (square). the focus is (square). the directrix is the line (square). the endpoints of the latus rectum are (square) and (square).

Explanation:

Step1: Identify the standard - form of the parabola

The standard form of a parabola opening left - right is $y^{2}=4px$. Given $y^{2}=-8x$, we can compare it with $y^{2}=4px$ to find $4p=-8$, so $p = - 2$.

Step2: Find the vertex

For a parabola of the form $y^{2}=4px$, the vertex is at the origin $(0,0)$.

Step3: Find the focus

The focus of a parabola $y^{2}=4px$ opening left - right is the point $(p,0)$. Since $p=-2$, the focus is $(-2,0)$.

Step4: Find the directrix

The directrix of a parabola $y^{2}=4px$ opening left - right is the line $x=-p$. Since $p = - 2$, the directrix is $x = 2$.

Step5: Find the endpoints of the latus rectum

The endpoints of the latus rectum of a parabola $y^{2}=4px$ are given by $(p,2p)$ and $(p, - 2p)$. Substituting $p=-2$, we get the endpoints $(-2,4)$ and $(-2,-4)$.

Answer:

The vertex is $(0,0)$.
The focus is $(-2,0)$.
The directrix is the line $x = 2$.
The endpoints of the latus rectum are $(-2,-4)$ and $(-2,4)$.