Problem 23 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=4$$

Step-by-Step Solution

Answer

The graph of the polar equation \(r = 4\) is a circle with a radius of 4 centered at the origin (0,0).

Step 1: Understand the Polar Equation

The polar equation given is \(r = 4\). It means that the distance from the pole (or origin) to any point on the graph is always 4.

Step 2: Sketch the Polar Graph

Since \(r = 4\) for all angles \(\theta\), this is a circle with a radius of 4 centered at the origin (0,0). Draw a circle on the polar plane with a radius of 4 units.

Step 3: Add Additional Points and Annotations

You can add points to the graph for clarity. For example, you might choose to plot points at \((4,0)\), \((0,4)\), \((-4,0)\), and \((0,-4)\). Also, it's useful to denote the radius of the circle on the graph.

Key Concepts

Graph Sketching

Sketching the graph of a polar equation begins with understanding the equation itself. In this case, the equation is simple, given by \( r = 4 \). Here, the variable \( r \) represents the radial distance from the pole (similar to the origin in cartesian coordinates) to any point on the graph.
This particular equation describes a circle where the radius is constantly 4 for any angle \( \theta \). This means that no matter which direction you face from the pole, you will always find yourself drawing the edge of this circle at a distance of 4 units away. Because of this consistent distance, sketching this graph involves drawing a uniform circle with a radius of 4. Utilize the polar grid, which consists of concentric circles centered at the origin, to ensure accuracy. These grids help visualize the distances and angles correctly and make sketching more precise. With the center of the circle at the pole, simply draw the circle touching the point \( 4 \) on the radial line in any direction.

Polar Coordinates

Polar coordinates are an alternative way of representing locations in a plane, particularly useful for circles and spirals. Rather than relying on the traditional \( (x, y) \) coordinate system, polar coordinates use \( (r, \theta) \). Here \( r \) signifies the distance from the pole, and \( \theta \) is the angle measured from the positive x-axis.
In the problem we are addressing, \( r \) is consistently 4, which indicates the points are exactly 4 units from the origin. This precision in determining radial distance is one of the benefits of using polar coordinates. When dealing with polar graphs, you depict the figure by understanding \( r \) in accordance with varying \( \theta \). However, since \( r \) is constant in this example, the angle \( \theta \) doesn't affect the size or position of the graph, only its direction from the center. This makes drawing a circle straightforward as the radius remains unchanging regardless of the angle.

Symmetry in Polar Graphs

Symmetry plays a significant role in simplifying the process of graphing polar equations. In polar coordinates, symmetry can occur about the pole, the line \( \theta = \frac{\pi}{2} \), or the polar axis (the positive x-axis). When an equation maintains its form while mirroring across one of these lines, it reveals symmetrical properties.
In the equation \( r = 4 \), symmetry simplifies graph sketching. Since the value of \( r \) is constant and independent of \( \theta \), the plot is symmetrical about all these lines. Such symmetry indicates a perfectly balanced shape—a circle in this case. This symmetry not only confirms accuracy in sketching but also aids in quick visualization. To determine symmetry in any polar graph, you can test it through substitution in the polar equation and examining if the equation holds true. Though such tests may not be explicitly needed for the given exercise, understanding the concept gives insight into more complex graphs.