6.3: Transformations of Polar Graphs
While playing around with your calculator one day, you create a polar plot that looks like this
Your teacher tells you that this is a polar plot with an equation
Can you find a way to do this by changing the equation you plot? Read on, and by the end of this Concept, you'll be able to do exactly that.
Watch This
Example of Graph of Polar Equation
Guidance
Just as in graphing on a rectangular grid, you can also graph polar equations on a polar grid. These equations may be simple or complex. To begin, you should try something simple like
Example A
On a polar plane, graph the equation
Solution: The solution is all ordered pairs of
Example B
On a polar plane, graph the equation
Solution: For this example, the
To begin graphing more complicated polar equations, we will make a table of values for
Example C
Graph the following polar equations on the same polar grid and compare the graphs.
Solution:
The cardioid is symmetrical about the positive
Changing the value of
It is also possible to create a sinusoidal curve called a limaçon. It has
As we've seen with cardioids, it is possible to create transformations of graphs of limaçons by changing values of constants in the equation of the shape.
Vocabulary
Cardioid: A carioid is a graph of two heart shaped loops reflected across the "x" axis.
Limacon: A limacon is a graph with a sinusoidal curve looping around the origin.
Transformation: A transformation is a change performed on a graph by changing the constants and/or the functions of the polar equations.
Guided Practice
1. Graph the curve
2. Graph the curve
3. Graph the curve
Solutions:
1.
2.
3.
Concept Problem Solution
As you've seen in this section, transformations to the graph of a cardioid can be accomplished by 2 different ways. In this case, you want to rotate the graph so that it is around the "x" axis instead of the "y" axis. To accomplish this, you change the function from a sine function to a cosine function:
Practice
Graph each equation.

r=4 
θ=60∘ 
r=2 
θ=110∘
Graph each function using your calculator and sketch on your paper.

r=3+3sin(θ) 
r=2+4sin(θ) 
r=1−5sin(θ) 
r=2−2sin(θ) 
r=3+6sin(θ) 
r=−3+6sin(θ)  Analyze the connections between the equations and their graphs above. Make a hypothesis about how to graph \begin{align*}r=a+b\sin(\theta )\end{align*} for positive or negative values of a and b where \begin{align*}b\geq a\end{align*}.
Graph each function using your calculator and sketch on your paper.
 \begin{align*}r=3+3\cos(\theta )\end{align*}
 \begin{align*}r=2+4\cos(\theta )\end{align*}
 \begin{align*}r=15\cos(\theta )\end{align*}
 \begin{align*}r=22\cos(\theta )\end{align*}
 \begin{align*}r=3+6\cos(\theta )\end{align*}
 \begin{align*}r=3+6\cos(\theta )\end{align*}
 Analyze the connections between the equations and their graphs above. Make a hypothesis about how to graph \begin{align*}r=a+b\cos(\theta )\end{align*} for positive or negative values of a and b where \begin{align*}b\geq a\end{align*}.
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Image Attributions
Here you'll learn how to alter graphs expressed in polar coordinates by changing the constants and/or functions used to describe the graph.