Problem 26 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=-3 \pi / 4$$

Step-by-Step Solution

Answer

The polar equation \(r=-3 \pi / 4\) represents a circle with a radius of \(3 \pi / 4\) units that's reversed (located in the opposite quadrant). The graph exhibits symmetry about the origin.

Step 1: Interpret the Polar Equation

In polar coordinates, the variable \(r\) represents the distance from the origin to the point and the angle \(\theta\) the measurement counterclockwise from the positive x-axis. Here, \(r=-3 \pi / 4\). This equation represents a circle with radius of \(3\pi/4\) units. However, because the radius is negative, this circle is reversed, meaning it lies in the quadrant opposite to what was initially considered. For this case, no angle is specified so it spans all possible angles.

Step 2: Identifying Symmetry

For this graph, there is symmetry about the origin, because for any point (r,θ) on the graph, the point (-r,θ+π) is also on the graph. This is because of the negative radius in the equation.

Step 3: Plotting the Graph

Now that we know what this polar equation represents, we can plot it. Remember that the circle's radius is \(3 \pi / 4\). Start by identifying the origin (0,0) then draw a circle with radius \(3\pi/4\) units in the opposite quadrant from what would be expected given the negative radius.

Key Concepts

Symmetry in Polar Graphs

Symmetry in polar graphs is an important feature to consider when sketching polar equations. Polar graphs can exhibit various types of symmetry: reflective concerning the x-axis, y-axis, or the origin. Detecting these symmetries helps to simplify the graphing process and understand the graph's shape and orientation more easily.

In the context of the given exercise, the polar equation for the circle with a negative radius, \( r = -\frac{3\pi}{4} \), introduces symmetry about the origin. This origin symmetry implies that for any point \((r, \theta)\) on the graph, the point \((-r, \theta + \pi)\) is also on the graph. This is because switching the positive radius to a negative one rotates the graph by 180 degrees.

Being aware of symmetry means you only need to plot a portion of the graph before you can predict or sketch the rest. If you identify symmetry early on, it saves time and ensures you don't miss parts of the graph.

Polar Equations

Polar equations express the relationship between the radius \( r \) and the angle \( \theta \) in polar coordinates. Unlike Cartesian equations that describe points on a plane via \( x \) and \( y \) coordinates, polar equations emphasize distance and direction.

A polar equation like \( r = -\frac{3\pi}{4} \) showcases how polar coordinates can sometimes behave unexpectedly compared to Cartesian ones. Here, the distance from the origin, given by \( r \), has a negative value. This negative value signifies that the point is located in the direction opposite to what is typically their expected location based on the angle \( \theta \). Essentially, it means the circle shifts its location by 180 degrees from its regular placement.

Understanding polar equations is crucial in converting and comparing different coordinate systems, particularly in cases like transforming the circular form in polar coordinates to a recognizable geometric shape.

Graphing Circles in Polar Coordinates

Graphing circles in polar coordinates introduces a unique way to view and understand these simple geometric shapes. In polar graphs, a circle centered at the origin with radius \( r \) has a straightforward representation, usually expressed as \( r = a \), where \( a \) is the radius.

However, when dealing with equations like \( r = -\frac{3\pi}{4} \), we recognize a circle with radius \( 3\pi/4 \), but due to the negative sign, it reflects into an opposite quadrant. Instead of simply sketching the circle outward from the origin, we acknowledge the negative sign causes a 180-degree shift in its typical position.

• Find the radius: Acknowledge the absolute value \( |r| \).
• Consider negative signs: Determine if any reflections occur, such as originating in an opposite quadrant.
• Visualize full rotation: Understand that in polar terms a negative radius plots in all directions given a full circle span \( 0 \leq \theta < 2\pi \).

Graphing circles with these considerations in mind provides a deeper understanding of how polar coordinates can manipulate simple shapes into different quadrants based on radius signs.