Polar Coordinates and Graphing

Objective To use polar coordinates. To graph polar equations. To graph special curves in polar coordinates

Polar Coordinates and Graphing r = directed distance = directed angle Polar Axis O Counterclockwise from polar axis to. Fixed point O is called the pole or origin

Plotting Points in the Polar Coordinate System The point lies two units from the pole on the terminal side of the angle. The point lies three units from the pole on the terminal side of the angle. The point coincides with the point. The point also is the same as the other two.

Multiple Representations of Points

Try plotting the following points in polar coordinates and find three additional polar representations of the point:

Converting from polar to rectangular The rectangular coordinates (x, y) and polar coordinates

Find the rectangular coordinates of each point with the given polar coordinates. Round to the nearest

Converting from rectangular to polar Polar coordinates of point P with rectangular coordinates (x, y) can be determined as follows:

Find polar coordinates of point A with rectangular coordinates

Graphing a Polar Equation circle

Using Symmetry to Sketch a Polar Graph Symmetry with respect to the line Symmetry with respect to the Polar Axis. Symmetry with respect to the Pole

Tests for Symmetry The graph of a polar equation is symmetric with respect to the following if the given substitution yields an equivalent equation.

Using Symmetry to Sketch Thus, the graph is symmetric with respect to the polar axis, and you need only plot points from 0 to . Lima  on

Symmetry Test Fails Unfortunately the tests for symmetry can guarantee symmetry, but there are polar curves that fail the test, yet still display symmetry. Let’s look at the graph of Try the symmetry tests. What happens? Original Replacement New Equation All of the tests indicate that no symmetry exists. Now, let’s look at the graph. \ Not symmetric about the polar axis. \ Not symmetric about the pole.

Spiral of Archimedes You can see that the graph is symmetric with respect to the line

Quick Test for Symmetry

Determine the effect of “a” on the graph of. As a gets larger, the graph gets bigger. Conversely as a Gets smaller the graph gets smaller.

What if a is negative? Reflects across the polar axis.

Determine the effect of “a” on the graph of. As a gets larger, the graph gets bigger. Conversely as a Gets smaller the graph gets smaller.

What if a is negative? Reflects across the line

Limacons General Equation What type of symmetry does the limacon have? What is the maximum value of r for each combination of the graph? What is the relationship between the negative values of r and the inner loop?

How does changing the value of “a” change the graph? What happens if “b” is an even number as it increases? What if “b” is odd? Explain the symmetry if the rose is a cosine equation versus a sine equation. The Rose

Summary of Special Polar Graphs Limacon with inner loop Limacons: Cardiod (heart-shaped) Dimpled Limacon Convex Limacon

Rose Curves:

Circles and Lemniscates CirclesLemniscates

Analyzing Polar Graphs Analyze the basic features of Type of Curve:Rose Curve with 2b petals = 4 petals Symmetry:Polar axis, pole, and Maximum Value of | r |:| r | = 3 when Zeros of r:r = 0 when You can use this same process to Analyze any polar graph.

Analyze and Sketch the graph: Type of curve: Symmetry: Maximum | r | Zeros of r Convex Limacon Cardioid Limacon polar axis none