Question

for the parabola:
$y^{2}=12x$
find the vertex and focus. make sure to enter your answers as coordinates
vertex:
$(0,0)$
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 parabola

The standard form of a parabola that opens to the right is \(y^{2}=4px\), where \((0,0)\) is the vertex, the focus is \((p,0)\) and the directrix is \(x = - p\).

Step2: Compare with the given equation

Given \(y^{2}=12x\), comparing with \(y^{2}=4px\), we have \(4p = 12\). Solving for \(p\), we get \(p=\frac{12}{4}=3\).

Step3: Find the focus

Since the focus of \(y^{2}=4px\) is \((p,0)\) and \(p = 3\), the focus is \((3,0)\).

Step4: Find the directrix

Since the directrix of \(y^{2}=4px\) is \(x=-p\) and \(p = 3\), the directrix is \(x=-3\).

Answer:

Focus: \((3,0)\)
Directrix: \(x = - 3\)