91Ó°ÊÓ

The focus of a parabola is a very special point that has a direct influence on its shape.
The standard form of a parabola that opens sideways, either to the left or right, is expressed as \((y-k)^2 = 4p(x-h)\). Here, the coordinates \((h, k)\) are the center of the parabola. The focus is located at \((h+p, k)\), where \(p\) is the distance from the vertex to the focus, which determines the width and direction of the parabola.

• If \(p\) is positive, the parabola opens to the right.
• If \(p\) is negative, it opens to the left.

To find the focus of the parabola given in the problem, \(y^2 = -12x\), we start by identifying \(4p = -12\). Solving for \(p\) gives us \(p = -3\). Substituting this value into the formula for the focus, \((h+p, k)\), results in the focus being at \((-3, 0)\).
This direct relationship helps guide how the parabola is graphed and its placement in a given coordinate system.

The directrix of a parabola is an essential component that works alongside the focus to define the parabola's shape. It is a straight line, unlike the curved line of the parabola itself.
For parabolas of the form \((y-k)^2 = 4p(x-h)\), the directrix is described by the equation \(x = h - p\). This means it is set \(p\) units away from the vertex, opposite the focus.

• If the parabola opens to the right with positive \(p\), the directrix will be to the left.
• Conversely, if the parabola opens to the left with negative \(p\), the directrix will be to the right.

Let's apply this to our given equation of the parabola, \(y^2 = -12x\). We found \(p = -3\) previously. Using the directrix formula \(x = h - p\) gives us \(x = 3\).
This line, \(x = 3\), provides a balance to the parabola's structure, ensuring that it maintains its defined shape and orientation.

The standard form of a parabola is crucial for understanding and graphing its equation and structure.
For parabolas that open left or right, the standard form is given by \((y-k)^2 = 4p(x-h)\). Here, \((h, k)\) is the vertex, the point around which the parabola is symmetrically aligned.
This form makes it easier to identify key components like:

• Vertex location.
• Direction the parabola opens.
• The values of \(p\), showing distance to the focus and the directrix.

The problem provides the equation \(y^2 = -12x\), which can be rewritten in this standard form by identifying parameters. Here, \(h = 0\) and \(k = 0\). With \(4p = -12\), solving for \(p\) gives us \(-3\). Consequently, \(y^2 = -12x\) aligns with \((y-0)^2 = 4(-3)(x-0)\).
Understanding this form allows you to quickly graph the parabola and locate its important features in any given problem.