Problem 50 - Precalculus

Question

Use a graphing utility to graph the polar equation. Describe your viewing window. $$r=-5 / 2$$

Step-by-Step Solution

Answer

The graph of this polar equation is a circle of radius 2.5 units left to the initial direction. The ideal viewing window on a graphing utility should be set to bear a circle of radius 2.5 units centered at the origin. Specifically, the window from \(x = -5\) to \(x = 5\), and \(y = -5\) to \(y = 5\), will suffice.

Step 1: Understand the Polar Coordinate System

Polar coordinates are a type of coordinate system where each point on a plane is determined by a distance from a reference point, called the origin, and an angle from a reference direction. The reference direction off which angles are measured is typically north, and the angle of any given measurement is often denoted by the Greek letter θ(Theta).

Step 2: Graph the equation

We are tasked with graphing the polar equation \(r = -5/2\). Since this equation does not involve theta, \(r\) is constant. This means that regardless of the theta value, \(r\) will be \(-5/2\) or \(-2.5\). The negative sign indicates that the graph will be a circle of radius 2.5 units, but in the opposite direction to the initial direction of measurement. In standard polar coordinates where the initial direction is off to the right, this circle will be on the left.

Step 3: Viewing window

The viewing window on your graphing utility should be set to include a circle of radius 2.5 units centered at the origin. Given that the circle will be on the left, as per the negative sign in the equation, a window from \(x = -5\) to \(x = 5\), and \(y = -5\) to \(y = 5\), should be sufficient to represent the graph.

Key Concepts

Polar Coordinate System

The polar coordinate system is a unique method for representing locations on a plane using two measures: distance and angle from a central point, which is known as the origin or pole. Unlike the more familiar Cartesian coordinates that use a grid of 'x' and 'y' values, polar coordinates express a point as \(r, \theta\), where \(r\) is the point's radial distance from the origin and \(\theta\) is the angle formed with the positive x-axis (to the right of the origin).

To visualize the polar coordinate system, imagine standing at the center of a circle and pointing directly to the right; this direction is your starting point, equivalent to 0 degrees. As you pivot in a counterclockwise direction, you measure angles from this reference line, which are represented in degrees or radians. The 'r' value can be positive or negative; a positive 'r' means moving outward from the origin in the direction of \(\theta\), while a negative 'r' suggests moving in the opposite direction. This feature makes polar coordinates particularly useful for representing cyclic patterns and circular shapes, as well as for solving problems involving angles and distances.

Polar Graph

A polar graph represents equations or data in the polar coordinate system, featuring a series of concentric circles that increase in radius from the point of origin. Each circle corresponds to a specific value of 'r', and lines emanating from the origin represent different angles \(\theta\). To plot the polar equation \(r = -5/2\), one would simply locate a point 2.5 units in the direction opposite the reference angle at every value of \(\theta\), because the equation tells us that the radial distance is constant and negative.

On a polar graph, the negative value of 'r' creates a distinctive pattern: although the angle \(\theta\) continues to rotate around the pole, all points are plotted 2.5 units in the reverse direction. This produces a circle centered on the origin but oriented in the opposite direction to the initial reference line. In our case, as the initial direction is usually to the right, reversing it would place the circle to the left. This behavior is a key aspect of polar graphs – the angle defines the direction of travel from the pole, while the magnitude and sign of 'r' define how far and in what way (towards or away from the reference angle) to move.

Graphing Utility Usage

Graphing utilities are invaluable tools for visualizing equations that may be complicated to draw by hand, especially in polar coordinates. When setting up to graph a polar equation like \(r = -5/2\), using a graphing utility, there are a few critical steps to ensure an accurate representation. Firstly, ensure the utility is set to 'polar mode', enabling you to input \(r\) and \(\theta\) values directly. Next, adjust your viewing window to capture the entire graph; in our example, extending the window to cover at least 5 units in each axis direction guarantees that the circle will be seen whole.

Additionally, consider the scaling of the radial lines and concentric circles, as they should be set such that increments of \(r\) and degrees or radians of \(\theta\) are represented clearly. Many utilities also allow you to trace the graph to see exact values at particular angles or to adjust the thickness and style of the graphed line for clarity. Through graphing utility usage, complex and abstract equations become tangible geometric representations, making them easier to understand and analyze.