Question

for the parabola:
$y^{2}=-20x$
find the vertex and focus. make sure to enter your answers as coordinates.
vertex:
focus:
write the equation of the directrix, and then sketch the parabola, the directrix and place a dot on the
vertex and the focus.
directrix:

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola of the form \(y^{2}=4ax\) has vertex \((0,0)\), focus \((a,0)\) and directrix \(x =-a\).
For the given parabola \(y^{2}=-20x\), we can rewrite it as \(y^{2}=4(- 5)x\).

Step2: Find the vertex

Comparing with the standard form \(y^{2}=4ax\), when \(x = 0\) and \(y = 0\), the vertex \((h,k)=(0,0)\).

Step3: Find the focus

Since \(4a=-20\), then \(a=-5\). The focus of the parabola \(y^{2}=4ax\) is \((a,0)\). Substituting \(a=-5\), the focus is \((-5,0)\).

Step4: Find the directrix

The directrix of the parabola \(y^{2}=4ax\) is \(x=-a\). Substituting \(a =-5\), we get \(x = 5\).

Answer:

Vertex: \((0,0)\)
Focus: \((-5,0)\)
Directrix: \(x = 5\)