Problem 18 - Precalculus: Mathematics for Calculus

Question

An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. $$y=\frac{1}{4} x^{2}$$

Step-by-Step Solution

Answer

Focus: (0, 1), Directrix: y = -1, Focal Diameter: 4.

Step 1: Rewrite the Equation in Standard Form

The given equation is \( y = \frac{1}{4} x^2 \). Since this equation fits the form \( y = ax^2 \), it is a vertical parabola centered at the origin. Comparing with the standard form \( y = \frac{1}{4p}x^2 \), we see that \( \frac{1}{4p} = \frac{1}{4} \). Hence, \( p = 1 \).

Step 2: Identify the Parameters

For a parabola of the form \( y = \frac{1}{4p} x^2 \), \( p \) is the distance from the vertex to the focus and from the vertex to the directrix along the y-axis. Here, \( p = 1 \). Thus, the focus is located at \( (0, p) = (0, 1) \), and the directrix is a horizontal line given by \( y = -p = -1 \). The focal diameter, or the length of the latus rectum, is \( 4p = 4 \).

Step 3: Graph the Parabola and the Directrix

To sketch the parabola, plot its vertex at the origin \((0, 0)\). The focus is \((0, 1)\), and the directrix is the line \( y = -1 \). The parabola opens upwards, with the distance from the vertex to the focus equaling the distance from the vertex to the directrix. The latus rectum provides the width of the parabola at the level of the focus, showing width \( 4 \) centered on the y-axis.

Key Concepts

Focus

The focus of a parabola is a key point that defines its shape and position. The basic idea is that every point on the parabola is equidistant from the focus and a corresponding point on the directrix, another core element of the parabola. This characteristic is what gives the parabola its distinctive "U" shape.

To find the focus of a vertical parabola represented by the equation such as \( y = \frac{1}{4p}x^2 \):

• The parameter \( p \) represents the distance from the vertex of the parabola to the focus.
• In our given example, \( p \) is determined by equating \( \frac{1}{4p} \) with \( \frac{1}{4} \), yielding \( p = 1 \).
• This means the focus for this particular parabola is at the point \( (0, 1) \).

Understanding the focus is crucial in graphing and comprehending how light or signals reflect off parabolic shapes, as they have practical applications in areas like acoustics and engineering.

Directrix

The directrix of a parabola is a fixed line used in the geometric definition. It plays a role in determining the set of distances required for a point to reside on the parabola. In simple terms, for every point on a parabola, the distance to the focus equals the distance to the directrix.

Depending on the orientation of the parabola, the directrix can be a horizontal or a vertical line.

• For a vertical parabola like \( y = \frac{1}{4} x^2 \), the directrix is always horizontal.
• In our example, because \( p = 1 \), the directrix is located at \( y = -p = -1 \).
• This implies the parabola and its directrix are symmetric about the vertex, making it an essential reference for sketching the graph.

This concept allows you to visualize how parabolas sit on a coordinate plane and helps with constructing accurate graphs, as well as solving more complex problems involving parabolas.

Focal Diameter

The focal diameter, also known as the latus rectum, is an integral feature of parabolas that describes the width at the focus level. This specific feature helps you to understand just how "wide" the parabola is at a certain point.

The formula for the focal diameter in a parabola is straightforward: it is equal to \( 4p \), where \( p \) is the distance from the vertex to the focus.

• In our specific example, where \( p = 1 \), the focal diameter becomes \( 4 \times 1 = 4 \).
• This means, at the level of the focus, the parabola stretches a total width of 4 units, centered on the y-axis.
• When sketching the parabola, this focal diameter provides a valuable reference to ensure the graph's proportionate representation.

The focal diameter's importance reflects in both mathematical theory and practical applications, such as designing satellites and communication devices, where focusing signals is crucial.