Problem 49 - Precalculus

Question

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=9 / 4$$

Step-by-Step Solution

Answer

The polar equation \( r = \frac{9}{4} \) graphs a circle centered at the origin with a radius of \( \frac{9}{4} \) units. The viewing window can be set from \( -3 \) to \( 3 \) on both axes to comfortably view the entire graph.

Step 1: Set the graph parameters in the graphing utility

Enter the polar equation \( r = \frac{9}{4} \) into the graphing utility's equation input field. Ensure that you're in the polar graphing mode. The viewing window should be set to view the entire circle comfortably. An appropriate viewing window would be \( x \) values ranging from \( -3 \) to \( 3 \) and \( y \) values also ranging from \( -3 \) to \( 3 \). This gives enough space to view the entire circle clearly.

Step 2: Generate the graph

Once the parameters are set, generate the graph using the graphing utility. It should plot a circle centered at the origin with a radius of \( \frac{9}{4} \) units.

Step 3: Describe the graph

The circle should be visible in its entirety in the viewing window, meaning you should be able to see a complete circle. The circle is centered at the origin, and every point on the circle is \( \frac{9}{4} \) units away from the origin.

Key Concepts

Understanding Graphing Utilities

Graphing utilities are powerful tools used to visualize mathematical equations, including polar equations. They can be software programs or even handheld devices that allow you to input equations and see their graphical representations. Specifically, a graphing utility in polar mode lets you explore curves like circles, spirals, and lemniscates, which are described using polar coordinates.

When working with a graphing utility:

• Make sure you set it to the correct mode—polar mode for polar equations.
• Input the equation accurately into the equation box provided by the utility.
• Adjust settings like viewing window so that you can view the graph properly.

By practicing these steps, you can graph various polar equations and examine their shapes and behavior in a visual format.

Polar Equations and Their Graphs

Polar equations are expressions that define a relationship between the radius, denoted as \( r \), and the angle \( \theta \) in the polar coordinate system. In contrast to Cartesian coordinates, where a point is defined by \( (x, y) \), polar coordinates define a point by how far away it is from the origin and at what angle from the positive x-axis.

The polar equation \( r = \frac{9}{4} \) signifies a circle centered at the origin. In this case:

• The radius \( r \) of the circle is constant, meaning every point on the circle is exactly \( \frac{9}{4} \) units from the origin.
• The equation is independent of \( \theta \), indicating that the radius is the same at all angles, which confirms the shape as a circle.

By inputting this equation into a graphing utility set to polar mode, you can observe its symmetrical nature and understand how polar coordinates map points onto a plane.

Choosing the Right Viewing Window

A viewing window is like an adjustable frame that allows you to see the graph of an equation clearly in the graphing utility. For polar equations like \( r = \frac{9}{4} \), choosing an appropriate viewing window is crucial to accurately analyzing the graph.

Here are tips for setting the right viewing window:

• Consider the radius of the polar graph. Since \( r = \frac{9}{4} \), the graph forms a circle with that radius; thus, set the x and y axes to range from \(-3\) to \(3\).
• This window size ensures the circle is displayed without being cut off or looking too small.
• Having a viewing window that fits your graph well will enhance your understanding by allowing you to see the full scope of the graph.

Ensuring you set an optimal viewing window adds to the effectiveness of your graphing utility and enhances your learning experience with polar graphs.