91Ó°ÊÓ

When dealing with parabolas, understanding the standard form of a parabola's equation is crucial for unlocking its characteristics. In a vertical parabola, the equation is usually written as \[ y = ax^2 + bx + c \]. However, for problems involving horizontal parabolas, like the given equation \( x = 3y^2 \), we rewrite it in the form\[ (y - k)^2 = 4p(x - h) \]. Here’s what each part signifies:

• \( h \) and \( k \) are the coordinates of the vertex, the point where the parabola changes direction.
• \( 4p \) gives us details about the parabola’s width and direction: \( p \) is the distance from the vertex to the directrix or focus.

A standard form equation allows for quick identification of a parabola’s key features, including its vertex and the direction in which it opens.
To rewrite \( x = 3y^2 \) in this form, as shown in the solution, you manipulate it into \( y^2 = \frac{1}{3}x \), making it easier to compare and find attributes like the vertex.

The directrix is an essential component of a parabola, helping to define its shape alongside the focus. Essentially, the directrix is a line that, together with the focus, determines every point on the parabola. Here’s why the directrix is important:

• A parabola is defined as the locus of points equidistant from the focus and the directrix.
• In a horizontal parabola, the directrix is a vertical line. For vertical parabolas, it is horizontal.

In the given equation \( x = 3y^2 \), we determined that the parabola opens towards the right. The vertex was found to be at \((0,0)\), leading to the identification of the directrix.
After finding \( p = \frac{1}{12} \), the equation of the directrix turned out to be \( x = -\frac{1}{12} \). This line lies to the left of the vertex, reflecting the parabola’s rightward opening.

Graphing a parabola on the coordinate plane provides a visual representation of its features, making analysis simpler and more intuitive. Here’s why understanding this concept is important when working with parabolas:

• The coordinate plane allows you to mark the vertex, directrix, and plot points through simple calculations.
• It aids in visualizing the symmetry and direction of the parabola, enhancing comprehension.

In our solution, the vertex at \((0,0)\) was plotted first. As the parabola opens to the right in this example, plotting additional points such as \( x = \frac{3}{4} \) with corresponding \( y \) values reveals the parabola's true shape.
The coordinate plane helps us see that the parabola is symmetric around the \( x \)-axis from the vertex, emphasizing its reflectivity and essence.