Question

Use a graphing utility and sketch the graph of \(r=\frac{6}{2 \sin \theta-3 \cos \theta}\).

Short answer

Use a graphing utility to plot the equation and observe the conic shape.

Step-by-step solution

Identify the Polar Equation

We are given the polar equation \(r = \frac{6}{2 \sin \theta - 3 \cos \theta}\). Notice it is expressed in terms of \(r\) and \(\theta\), which are common in polar coordinates.

Simplify and Set Up for Graphing

Examine the polar equation \(r = \frac{6}{2 \sin \theta - 3 \cos \theta}\) and recognize that it can represent a conic section. Simplify if necessary, but this equation is already in a form where \(r\) is explicitly solved.

Use a Graphing Utility

Open a graphing utility capable of plotting polar coordinates. Examples of such tools include Desmos, GeoGebra, or any graphing calculator with a polar mode.

Enter the Polar Equation

In the graphing utility, enter the given polar equation \(r = \frac{6}{2 \sin \theta - 3 \cos \theta}\) exactly as it is given. Ensure the input recognizes \(\sin(\theta)\) and \(\cos(\theta)\).

Adjust Graphing Settings

Check and set the domain for \(\theta\), typically from \(0\) to \(2\pi\), to ensure a complete graph of the polar equation is generated. Adjust the scale or range if necessary for clarity.

Sketch the Graph

Once the graph is plotted, sketching it involves capturing the shape and key features as displayed by the utility. This graph will show how \(r\) changes with \(\theta\).

Interpret the Graph

Observe the graphed image to understand its shape and symmetry, which often provides insights into the nature of the polar equation. This particular plot will likely represent a conic section, such as a hyperbola, as suggested by its general form.

Key concepts

graphing utility

A graphing utility is a valuable tool for visualizing mathematical equations, especially those in complex forms like polar coordinates. These utilities can handle various types of equations, helping you understand and analyze their behavior visually.

To use a graphing utility effectively, follow these steps:

• Choose a tool with polar graphing capabilities such as Desmos or GeoGebra.
• Familiarize yourself with the interface, especially how to input polar equations accurately.
• Set your angles, typically from 0 to \(2\pi\), to capture the full graph.
• Explore options for resizing and adjusting the plot for better visibility and interpretation.

Graphing utilities not only show you the graph but also allow you to interact with the equation—changing parameters and seeing instant visual results. This hands-on approach makes understanding complex equations much easier.

conic sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These sections include ellipses, parabolas, hyperbolas, and circles. In polar form, these curves take on unique representations that can be identified by their general equation structure.

Recognizing conic sections in polar coordinates requires experience with their forms. A polar equation such as \(r = \frac{6}{2 \sin \theta - 3 \cos \theta}\), for example, indicates a type of conic. The constant terms and the angle functions reveal clues about the shape:

• Ellipses often appear with terms involving both \(\sin\) and \(\cos\).
• Hyperbolas can have larger dominating terms, indicating openings or asymptotes.
• Parabolas might appear when one trigonometric term is missing.
• Circles are usually simple and involve \(\cos(\theta)\) or \(\sin(\theta)\) directly.

Using graphing utilities, you can directly visualize these structures for clearer understanding.

polar equation

A polar equation is based on the polar coordinate system, which describes a point in terms of a radius \(r\) and an angle \(\theta\). This system offers a unique way to represent relationships, especially those involving rotational symmetry or circular motion.

When dealing with polar equations, keep these in mind:

• The angle \(\theta\) is typically measured from the positive x-axis around the origin.
• \(r\) can be positive or negative, affecting the direction relative to the origin.
• Polar forms simplify the calculation of symmetries and angles, often revealing insights hidden in Cartesian form.

Understanding and interpreting polar equations involves converting between polar and Cartesian (rectangular) forms. This conversion aids in solving specific problems, such as intersections with lines or other curves, which is crucial in advanced geometry.