Problem 24 - Precalculus with Limits

Question

In Exercises 23-48, sketch the graph of the polar equation using symmetry, zeros, maximum \(r\)-values, and any other additional points. \(r=-7\)

Step-by-Step Solution

Answer

The graph for the polar equation \(r=-7\) is a circle with a radius of \(7\) units, reflected across the origin. It lies in the darker region, opposite to the brighter region where a circle for positive \(r\) values would be plotted.

Step 1: Understand the equation

First, understand that the equation \(r=-7\) is a simple constant. Normally, the polar equation of a circle centered at the origin is \(r=a\), where \(a\) is the radius of the circle. However, in our case, the radius is negative. This negative sign indicates that we'll plot our circle not in the brighter region (where the radius extends towards), but we'll reflect it to the darker side. Taking these points into consideration, we proceed to the actual graph drawing.

Step 2: Sketch the graph

Take a polar graph paper. Locate the origin point, where \(r=0\). The radius of the circle should be \(7\) units due to the absolute value of \(r=-7\). Therefore, plot a circle with a radius of \(7\) units. However, instead of plotting on the brighter side as we would for positive \(r\) values, we reflect it to the darker side.

Step 3: Interpret the graph

The graph represents a circle with a radius of \(7\), reflected across the origin. It's centered at the origin, and doesn't lie in the brighter region due to its negative radius. All points on this circle satisfy the given equation \(r=-7\) in the polar coordinate system.

Key Concepts

Polar Coordinate System

Understanding the polar coordinate system is crucial for mastering the art of graphing equations that define shapes such as circles, spirals, and roses. Unlike the Cartesian coordinate system, which uses horizontal and vertical axes (x and y) to locate a point in a plane, the polar coordinate system identifies a point based on its distance from a reference point called the pole (the origin in Cartesian terms) and the angle from a reference direction (usually the positive x-axis).

A point in the polar coordinate system is expressed as \(r, \theta\), where \(r\) is the radial distance from the pole, and \(\theta\) is the angular coordinate, typically measured in degrees or radians from the reference direction. It's important to note that a negative \(r\) value indicates that the point is located in the opposite direction from the pole at \(r\) units away, which can often be confusing for beginners.

Graphing Polar Equations

The process of graphing polar equations involves plotting points and understanding how the equation corresponds to a shape. For example, a simple polar equation like \(r = a\) represents a circle with a radius \(a\) centered at the pole. More complex equations can represent spirals, roses, or limaçons.

To sketch the graph of a polar equation, start by determining important features such as symmetry, zeros (where \(r=0\)), and maximum \(r\)-values. Use polar graph paper which comes with concentric circles and radial lines to facilitate accurate plotting. It's essential to recognize that the same point can have multiple polar representations due to the periodic nature of the angular coordinate \(\theta\).

For the exercise involving \(r=-7\), one plots a circle with radius 7 units. Understanding the negative radius is key; it indicates the circle is reflected across the pole to the opposite side. This reflection is a unique feature of polar coordinates that might not have an analogue in the Cartesian system.

Symmetry in Polar Graphs

Symmetry in polar graphs can greatly simplify the graphing process. There are three types of symmetry to consider: symmetry with respect to the line \(\theta = 0\) (the polar axis), symmetry with respect to the line \(\theta = \frac{\pi}{2}\), and symmetry with respect to the pole (the origin).

If a polar equation remains unchanged when \(\theta\) is replaced with \(\theta + \pi\), that indicates symmetry about the polar axis. If replacing \(r\) with -\(r\) doesn't change the equation, the graph is symmetric with respect to the pole. These symmetries suggest that one only needs to plot half or a quarter of the points to complete the whole graph since the rest can be filled in by reflecting across the lines of symmetry.

For the given equation \(r=-7\), it inherently involves reflection across the pole due to the negative radius. All points on its graph will mirror across the origin, effectively showing polarity symmetry. When graphing this equation, one needn't plot multiple values since the symmetry dictates the final shape – a circle, in this case.