Question

complete each sentence with the appropriate properties of the given parabola.

$y^{2}=12x$

the value of p is

the equation of the directrix is

the focus is located at

the parabola is concave

Explanation:

Step1: Recall the standard form of parabola

The standard form of a parabola \(y^{2}=4px\).

Step2: Compare with 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 equation of the directrix

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

Step4: Locate the focus

For the parabola \(y^{2}=4px\), the focus is at \((p,0)\). Substituting \(p = 3\), the focus is at \((3,0)\).

Step5: Determine the concavity

For the parabola \(y^{2}=4px\), when \(p>0\), the parabola is concave to the right. Here \(p = 3>0\).

Answer:

The value of \(p\) is \(3\).
The equation of the directrix is \(x = - 3\).
The focus is located at \((3,0)\).
The parabola is concave to the right.