To determine the equation of the directrix for the given parabola [tex]\( y^2 = 12x \)[/tex], follow these steps:
1. Identify the Standard Form:
The standard form for a horizontally-opening parabola is [tex]\( y^2 = 4ax \)[/tex], where [tex]\( a \)[/tex] is the distance from the vertex to the focus.
2. Compare with the Given Equation:
The given equation is [tex]\( y^2 = 12x \)[/tex].
To match it with the standard form [tex]\( y^2 = 4ax \)[/tex], we need to find [tex]\( a \)[/tex] such that [tex]\( 4a = 12 \)[/tex].
3. Solve for [tex]\( a \)[/tex]:
[tex]\[
4a = 12
\][/tex]
[tex]\[
a = \frac{12}{4} = 3
\][/tex]
4. Determine the Directrix:
For a horizontally-opening parabola [tex]\( y^2 = 4ax \)[/tex], the directrix is given by [tex]\( x = -a \)[/tex].
Here, [tex]\( a \)[/tex] is 3, so the directrix is:
[tex]\[
x = -3
\][/tex]
Therefore, the equation representing the directrix of the parabola [tex]\( y^2 = 12x \)[/tex] is [tex]\( \boxed{x = -3} \)[/tex].
