Question

Identify the focus, directrix, and axis of symmetry of the parabola. Graph the equation. (See Example 2.) $$x=\frac{1}{24} y^2$$

Short answer

The focus of the parabola is at (1,0), the directrix is the line \(x = -1\), and the axis of symmetry is the y-axis.

Step-by-step solution

Identify the coefficient a

In the equation \(x = \frac{1}{24} y^2\), the coefficient \(a\) represents \(\frac{1}{24}\).

Determine the focus

Recall that the focus lies on the line \(x = \frac{1}{4a}\). Substituting the value of \(a\) into this equation provides the focus, which is at \(x = \frac{1}{4* \frac{1}{24}}\), leading to the coordinate (1,0).

Determine the directrix

Recall that the directrix of the parabola is the line \(x = -\frac{1}{4a}\). Substituting the value of \(a\) into this equation gives the equation of directrix as \(x = -\frac{1}{4* \frac{1}{24}}\), which simplifies to \(x = -1\).

Identify the axis of symmetry

In parabolas of the form \(x = ay^2\), the axis of symmetry is always the y-axis. Therefore, in this case, the axis of symmetry is the line \(y=0\).

Graph the parabola

Plot the points known from the focus, directrix and axis of symmetry, and use these to draw the graph. Note that the parabola opens to the right, as \(a > 0\).

Key concepts

Focus of a Parabola

The focus of a parabola is a significant point that plays a crucial role in its geometric structure. The focus is a fixed point located inside the parabola, and it has the unique property that the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix.
For the equation given, the parabola can be expressed as \( x = rac{1}{24}y^2 \). This is a standard form where the axis of symmetry is vertical. To find the focus, use the formula \( x = \frac{1}{4a} \).
In this example, the coefficient \( a \) is \( \frac{1}{24} \), so substituting it in gives us \( x = \frac{1}{4 \times \frac{1}{24}} \), simplifying to \( x = 1 \). Thus, the focus of this parabola is located at the point \( (1, 0) \).

Directrix of a Parabola

The directrix of a parabola is a line that is equally important as the focus. It is not a point, but rather a line that serves as a reference point for measuring distances from the parabola. Every point on the parabola is equidistant from the focus and this directrix line.
For horizontal parabolas like \( x = rac{1}{24}y^2 \), the directrix can be determined using the formula \( x = -\frac{1}{4a} \). With the value of \( a \) being \( \frac{1}{24} \), the directrix line is found by calculating \( x = -\frac{1}{4 \times \frac{1}{24}} \), which simplifies to \( x = -1 \). The directrix, then, is the vertical line \( x = -1 \).

Axis of Symmetry

The axis of symmetry in a parabola is an imaginary line that divides the parabola into two mirror-image halves. For any parabola, this line always passes through the vertex, which is the "turning point" of the parabola.
In the general form of the equation \( x = ay^2 \), the axis of symmetry is straightforwardly the line \( y = 0 \), or the y-axis. This is because the parabola is oriented horizontally.
The role of this axis is to help in accurately sketching the parabola and understanding its symmetry. For our equation, since everything is centered around the y-axis, every point on one side has a symmetric counterpart on the other side.

Graphing Parabolas

When graphing a parabola like \( x = rac{1}{24}y^2 \), understanding its components – focus, directrix, and axis of symmetry – will guide you in the plot. First, locate the focus at \( (1, 0) \) and draw the directrix line at \( x = -1 \).
With the axis of symmetry on the y-axis \( y = 0 \), plot a few key points by choosing \( y \) values. Calculate the corresponding \( x \) values using the equation to sketch the general shape.
In this equation, since \( a > 0 \), the parabola opens to the right. It appears as a wide "U" shape curving outward from the axis of symmetry. Understanding these aspects allows you to visualize the shape and direction of the parabola effectively.