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Graphing Polar Equations

Author: Sophia
PDF Version

what's covered
In this lesson, you will graph equations using their polar forms. Specifically, this lesson will cover:

Table of Contents

1. Graphing a Polar Equation by Using Its Rectangular Form

Consider the graphs of x squared plus y squared equals 4 in rectangular coordinates and r equals 2 in polar coordinates.

Their graphs on their respective coordinate systems are shown below.

bold italic x to the power of bold 2 bold plus bold italic y to the power of bold 2 bold equals bold 4 bold italic r bold equals bold 2
A graph with an x-axis and a y-axis ranging from −3 to 3. A circle is centered at (0, 0) and passes through the points (2, 0), (0, 2), (−2, 0), and (0, −2). A polar coordinate graph with a horizontal and a vertical line intersecting at the origin. A circle with radius 2 is traced.

The graphs are identical; they are just plotted in two different coordinate systems.

One way to graph polar equations is to transform the polar equation into a rectangular equation, then graph the rectangular equation. This is useful since we are more familiar with rectangular equations and their graphs.

EXAMPLE

Consider the polar equation r equals 6   cos   theta.

First, transform the equation to rectangular coordinates.

r equals 6   cos   theta This is the original equation.
r squared equals 6 r   cos   theta Multiply both sides of the equation by r.
x squared plus y squared equals 6 x Replace r squared with x squared plus y squared and r   cos   theta with x.
x squared minus 6 x plus y squared equals 0 Subtract 6 x from both sides and place it after the x squared term.

try it
Consider the polar equation r equals fraction numerator 6 over denominator cos   theta minus 2   sin   theta end fraction.
Transform this equation into a rectangular equation.
r equals fraction numerator 6 over denominator cos   theta minus 2   sin   theta end fraction The original equation.
r open parentheses cos   theta minus 2   sin   theta close parentheses equals 6 Multiply both sides by the denominator.
r   cos   theta minus 2 r   sin   theta equals 6 Distribute r.
x minus 2 y equals 6 Replace r   cos   theta with x and r   sin   theta with y. This equation is now written in rectangular form.
Describe the graph of the equation.
The equation produces a line with x-intercept open parentheses 6 comma space 0 close parentheses and y-intercept open parentheses 0 comma space short dash 3 close parentheses.

2. Testing Polar Equations for Symmetry

It is not always convenient to find the corresponding rectangular equation given a polar equation; therefore, other properties of graphs are needed. One of them is symmetry. If a graph has a specific type of symmetry, then we are able to obtain the entire graph by sketching half of it.

Consider the following figure, which shows four key points on a circle that is centered at the origin, the polar axis, and the line theta equals straight pi over 2.

A polar coordinate graph with a horizontal axis and a vertical axis intersecting at the origin. The horizontal axis is labeled ‘Polar axis’, and the vertical axis is labeled theta equals pi over 2. The graph has concentric circles and radial lines forming a circular grid. A circle is highlighted and centered at the origin. A dashed line through the origin connects (r, theta) in quadrant 1 and (-r, theta) in quadrant 3, and another dashed line through the origin connects the points ((-r, -theta) in quadrant 2 and (r, -theta) in quadrant 4.

This graph exhibits three types of symmetry:

Type of Symmetry Condition for Symmetry
Symmetry about the polar axis (the x-axis) Given open parentheses r comma space theta close parentheses is a point on the curve, then open parentheses r comma space short dash theta close parentheses is also a point on the curve.
Symmetry about the line theta equals straight pi over 2 (the y-axis) Given open parentheses r comma space theta close parentheses is a point on the curve, then open parentheses short dash r comma space short dash theta close parentheses is also a point on the curve.
Symmetry about the pole (origin) Given open parentheses r comma space theta close parentheses is a point on the curve, then open parentheses short dash r comma space theta close parentheses is also a point on the curve.

The most efficient way to test for symmetry is to replace open parentheses r comma space theta close parentheses with open parentheses r comma space short dash theta close parentheses for polar axis symmetry, open parentheses short dash r comma space short dash theta close parentheses for symmetry about the line theta equals straight pi over 2 comma and open parentheses short dash r comma space theta close parentheses for symmetry about the pole.

Note that these tests guarantee symmetry. If a symmetry test fails, then the corresponding graph could still have that type of symmetry. This happens because there are several ways to express the same point using polar coordinates, and checking all possible substitutions would be time consuming. Some types of symmetry can still be identified by plotting points that reflect around the axes or the pole, as we will see later.

EXAMPLE

Test the equation r equals 2   cos   theta for symmetry.

Symmetry about the polar axis:

r equals 2   cos open parentheses short dash theta close parentheses Replace theta with short dash theta and keep r the same.
r equals 2   cos   theta Use an even/odd identity.

Since this is the original equation, r equals 2   cos   theta has symmetry about the polar axis.

Symmetry about the line theta equals straight pi over 2 colon

short dash r equals 2   cos open parentheses short dash theta close parentheses Replace r with -r and theta with short dash theta.
short dash r equals 2   cos   theta Use an even/odd identity on the right side.
r equals short dash 2   cos   theta Solve the equation for r.

Since this is not the original equation, r equals 2   cos   theta fails the test for symmetry about the line theta equals straight pi over 2.

Symmetry about the pole:

short dash r equals 2   cos   theta Replace r with -r and keep theta the same.

Since this is not the original equation, r equals 2   cos   theta fails the test for symmetry about the pole.

It can be shown that the rectangular form of the equation r equals 2   cos   theta is open parentheses x minus 1 close parentheses squared plus y squared equals 1 comma which is a circle with center open parentheses 1 comma space 0 close parentheses and radius 1. The graph of the polar equation is shown below, which confirms symmetry about the polar axis and shows that it does not have the other types of symmetry.

The graph of a circle that passes through the origin, (2, 0), (1, 1) and (1, -1).

try it
Consider the polar equation r equals 4   sin open parentheses 3 theta close parentheses. In this exercise, you will will test the equation for symmetry about the polar axis, the line theta equals straight pi over 2 comma and the pole.
Test the equation for each type of symmetry. Which test(s) pass and which test(s) fail?
Note that we will make use of the identity sin open parentheses short dash theta close parentheses equals short dash sin   theta comma which is an even/odd identity.

To test for symmetry about the polar axis, replace open parentheses r comma space theta close parentheses with open parentheses r comma space short dash theta close parentheses comma then determine if the equation is equivalent.

Making the replacement, we have r equals 4   sin open parentheses 3 open parentheses short dash theta close parentheses close parentheses equals 4   sin open parentheses short dash 3 theta close parentheses equals short dash 4   sin open parentheses 3 theta close parentheses. This equation is not identical, so the test fails for polar axis symmetry.

To test for symmetry about the line theta equals straight pi over 2 comma replace open parentheses r comma space theta close parentheses with open parentheses short dash r comma space short dash theta close parentheses comma then determine if the equation is equivalent.

Making the replacement, we have:

short dash r equals 4   sin open parentheses 3 open parentheses short dash theta close parentheses close parentheses short dash r equals 4   sin open parentheses short dash 3 theta close parentheses short dash r equals short dash 4   sin open parentheses 3 theta close parentheses space space space r equals 4   sin open parentheses 3 theta close parentheses

Since this is the same as the starting equation, the graph has symmetry about the line theta equals straight pi over 2.

To test for symmetry about the pole, replace open parentheses r comma space theta close parentheses with open parentheses short dash r comma space theta close parentheses comma then determine if the equation is equivalent.

Making the replacement, we have short dash r equals 4   sin open parentheses 3 theta close parentheses.

Now, solve for r: r equals short dash 4   sin open parentheses 3 theta close parentheses.

This equation is not identical, so the test fails for symmetry about the pole.

In conclusion, only symmetry about the line theta equals straight pi over 2 is guaranteed.

3. Finding Zeros and Maximum Values of |  r   |

Recall that r equals 0 represents a single point, the pole (origin).

Given a polar equation r equals f open parentheses theta close parentheses comma the solutions of the equation f open parentheses theta close parentheses equals 0 represent points on the graph that pass through the origin.

EXAMPLE

Consider the equation r equals 1 plus 2   sin   theta. Find all zeros on the interval open square brackets 0 comma space 2 straight pi close parentheses.

r equals 1 plus 2   sin   theta This is the original equation.
0 equals 1 plus 2   sin   theta The zeros occur when r equals 0.
sin   theta equals short dash 1 half Isolate sin   theta on one side.
theta equals fraction numerator 7 straight pi over denominator 6 end fraction comma space fraction numerator 11 straight pi over denominator 6 end fraction Find all values of theta on the interval open square brackets 0 comma space 2 straight pi close parentheses such that the y-coordinate of the point is short dash 1 half.

Thus, the zeros occur when theta equals fraction numerator 7 straight pi over denominator 6 end fraction comma space fraction numerator 11 straight pi over denominator 6 end fraction.

try it
Consider the polar equation r equals 2 plus 2   cos   theta.
Find all zeros on the interval [0, 2π).
To find the zeros, set r equals 0 and solve:

0 equals 2 plus 2   cos   theta Replace r with 0.
short dash 2 equals 2   cos   theta Subtract 2 from both sides.
short dash 1 equals cos   theta Divide both sides by 2.
theta equals straight pi According to the unit circle, the only solution is theta equals straight pi.

Therefore, theta equals straight pi is the only zero on the interval open square brackets 0 comma space 2 straight pi close parentheses.

Recall that sine and cosine functions have minimum and maximum values. Therefore, the points where the largest and smallest values of open vertical bar r close vertical bar occur can be found by analyzing the expression.

EXAMPLE

Consider the equation r equals 2 plus 2   sin   theta. Find all points where the maximum value of open vertical bar r close vertical bar occurs.

The quantity 2   sin   theta has a maximum value of 2, which means r equals 2 plus 2   sin   theta has a maximum value of 4. Thus, the maximum value of open vertical bar r close vertical bar is 4.

Next, find all points where the graph attains the maximum value of open vertical bar r close vertical bar comma which is r equals 4.

r equals 2 plus 2   sin   theta This is the original equation.
4 equals 2 plus 2   sin   theta Replace r by its maximum value.
sin   theta equals 1 Isolate sin   theta to one side.
theta equals straight pi over 2 Solve for theta.

Therefore, the polar point open parentheses r comma space theta close parentheses equals open parentheses 4 comma space straight pi over 2 close parentheses is the location of the point with the maximum value of r.

try it
Consider the equation r equals 5 minus 3   cos   theta.
Find all polar coordinates where |  r   | attains its maximum value.
The quantity short dash 3   cos   theta has maximum value 3, which means that r equals 5 minus 3   cos   theta has maximum value 5 plus 3 equals 8.

To find all points where the maximum value occurs, set r equals 8 and solve.

8 equals 5 minus 3   cos   theta Set r equals 8.
3 equals short dash 3   cos   theta Subtract 5 from both sides.
short dash 1 equals   cos   theta Divide both sides by -3.
theta equals straight pi From the unit circle, theta equals straight pi.

The location of the maximum value of open vertical bar r close vertical bar is open parentheses 8 comma space straight pi close parentheses.

4. Graphing Polar Equations

4a. Forming Graphs by Plotting Points

Taking the information we found so far, and adding some other solution points to a polar curve, we can graph the polar curve.

EXAMPLE

Sketch the graph of r equals 2 plus 2   cos   theta.

Through previous examples, we know that the graph has symmetry about the polar axis, passes through the pole at the point open parentheses 0 comma space straight pi close parentheses comma and the maximum value of open vertical bar r close vertical bar is attained at the point open parentheses 4 comma space 0 close parentheses.

Since the graph is symmetric with respect to the polar axis, we only need to plot points between theta equals 0 and theta equals straight pi.

Here is a table of values for special angles between theta equals 0 and theta equals straight pi. Note that non-exact values are rounded to two decimal places.

bold italic theta 0 straight pi over 6 straight pi over 4 straight pi over 3 straight pi over 2 fraction numerator 2 straight pi over denominator 3 end fraction fraction numerator 3 straight pi over denominator 4 end fraction fraction numerator 5 straight pi over denominator 6 end fraction straight pi
bold italic r bold equals bold 2 bold plus bold 2 bold   bold cos bold   bold italic theta 4 3.73 3.14 3 2 1 0.59 0.27 0

The complete graph is shown below.

A graph in the shape of a lima bean with a pointed dimple, on its side so that the dimple is on the left. The graph starts 4 units to the right of the origin, curves inward as it moves counterclockwise, passing through a point 2 units above the origin, then continuing inward until reaching the cusp at the origin. Then it curves outward as it continues counterclockwise, eventually curving out to the starting point.

A graph with this shape is called a cardioid, which resembles a heart. Notice the cusp on the left side of the curve. Depending on the equation, the cusp could appear at the top, bottom, left, or right side of the curve. The general equations of cardioids are discussed later.

EXAMPLE

Consider the equation r equals 3 plus 2   sin   theta.

By performing the symmetry tests, none of them can confirm that this graph has any type of symmetry.

The minimum value of r is 1, while the maximum value of r is 5. This means that there is no point where the graph intersects the pole.

Shown below is the graph of the equation.

The graph of a cardioid which resembles a circle with a dimple at its base. In a counterclockwise direction, the graph starts out 3 units to the right of the origin, curves out to a point that is 5 units above the origin, then back to a point that is 3 units left of the origin, then to a point 1 unit below the origin, then back to its starting point.

Notice that even though the test failed for symmetry about the line theta equals straight pi over 2, this graph is symmetric about the line theta equals straight pi over 2.

This graph is called a limacon, which resembles a lima bean. Notice the “dimple” at the bottom of the curve. Depending on the equation, the dimple could occur at the top, bottom, left, or right side of the curve. The general equations of limacons are discussed later.

watch
To see the process of graphing a rose curve (or petal curve) described below, click on this video.

4b. Graphs of Classic Polar Equations

By applying the techniques we have learned so far, we could graph many different types of polar equations. The graphs shown below represent the equations that are easier to graph using polar coordinates than their rectangular counterparts.

Even though these could be transformed to rectangular coordinates quite easily, the polar equations of circles centered on an axis are more convenient.

Their graphs are shown below.

The graph of a circle that has leftmost point at the origin and rightmost point some distance to the right. The equation of the graph is r equals a cos theta with a positive. The graph of a circle that has rightmost point at the origin and leftmost point some distance to the left. The equation of the graph is r equals a cos theta with a negative.
The graph of a circle that has lowest point at the origin and highest point some distance above the origin. The equation of the graph is r equals a sin theta with a positive. The graph of a circle that has highest point at the origin and lowest point some distance below the origin. The equation of the graph is r equals a sin theta with a negative.

Next, let’s look at polar equations of the form r equals a plus-or-minus b   cos   theta and r equals a plus-or-minus b   sin   theta comma where a greater than 0 and b greater than 0.

Cardioids, where a equals b.

A graph in the shape of a lima bean with a pointed dimple, on its side so that the dimple is on the left. The graph starts at a point to the right of the origin, curves inward as it moves counterclockwise, touches the origin, then curves outward as it continues counterclockwise, eventually curving out to the starting point. A graph in the shape of a lima bean with a pointed dimple, on its side so that the dimple is on the right. The graph starts at the origin at a cusp, curves outward as it moves counterclockwise, reaches its furthest point left of the origin, then curves inward as it continues counterclockwise, reaching its starting point.
A graph in the shape of a lima bean with a pointed dimple, upside down that the dimple is on the bottom. The highest point on the graph is above the origin, and moving counterclockwise, the graph curves inward, crossing the second and third quadrants, reaching the cusp at the origin. At this point, the graph curves outward in the same fashion, crossing through the fourth and first quadrants, reaching its starting point. A graph in the shape of a lima bean with a pointed dimple, oriented so that the dimple is on the top. The lowest point on the graph is below the origin, and moving counterclockwise, the graph curves inward, crossing the fourth and first quadrants, reaching the cusp at the origin. At this point, the graph curves outward in the same fashion, crossing through the second and third quadrants, reaching its starting point.

Limacons with dimples, where a greater than b.

A closed curve called a limacon, resembling a circle with a dimple on its left side, 1 unit to the left of the origin and its rightmost point is further to the right of the origin. The curve represents the equation r equals a + b cos theta. A closed curve called a limacon, resembling a circle with a dimple on its right side, 1 unit to the left of the origin and its leftmost point is further to the right of the origin. The curve represents the equation r equals a - b cos theta.
A closed curve called a limacon, resembling a circle with a dimple on its base, 1 unit below the origin and its highest point is more than 1 unit above the origin. The curve represents the equation r equals a + b sin theta. A closed curve called a limacon, resembling a circle with a dimple on its base, 1 unit above the origin and its lowest point is more than 1 unit below the origin. The curve represents the equation r equals a - b sin theta.

Limacons with inner loops, where a less than b.

The graph of a limacon with an inner loop. The graph is symmetric over the positive x-axis, with the loop having endpoints at the origin and 2 units to the right of the origin. The graph of a limacon with an inner loop. The graph is symmetric over the negative x-axis, with the loop having endpoints at the origin and 2 units to the left of the origin.
The graph of a limacon with an inner loop. The graph is symmetric over the y-axis, with the loop having vertical endpoints at (0, 0) and 2 units above the origin. The graph of a limacon with an inner loop. The graph is symmetric over the y-axis, with the loop having vertical endpoints at (0, 0) and 2 units below the origin.

Another type of polar graph looks like the “infinity” symbol. These are called lemniscates, whose equations are r squared equals a squared cos open parentheses 2 theta close parentheses and r squared equals a squared sin open parentheses 2 theta close parentheses comma where a not equal to 0.

A polar coordinate graph with an x-axis (polar axis) and a y-axis intersecting at the origin. The graph contains concentric circles and radial lines spaced at regular angular intervals. A lemniscate-shaped curve is centered at the origin, extending horizontally along the x-axis. The curve has two equal lobes, one on each side of the origin. The graph represents the equation r squared equals a squared cos(2 theta). A polar coordinate graph with an x-axis (polar axis) and a y-axis intersecting at the origin. The graph contains concentric circles and radial lines spaced at regular angular intervals. A lemniscate-shaped curve is centered at the origin, extending vertically along the y-axis. The curve has two equal lobes, one on each side of the origin. The graph represents the equation r squared equals −a squared cos(2 theta).
A polar coordinate graph with an x-axis (polar axis) and a y-axis intersecting at the origin. The graph contains concentric circles and radial lines spaced at regular angular intervals. A lemniscate-shaped curve is centered at the origin and extends diagonally, with two symmetrical lobes—one in the first quadrant and the other in the third quadrant. The graph represents the equation r squared equals a squared sin(2theta). A polar coordinate graph with an x-axis (polar axis) and a y-axis intersecting at the origin. The graph contains concentric circles and radial lines spaced at regular angular intervals. A lemniscate-shaped curve is centered at the origin and extends diagonally, with two symmetrical lobes—one in the second quadrant and the other in the fourth quadrant. The graph represents the equation r squared equals −a squared  sin(2*theta).

A rose curve (or petal curve) is formed by the equations r equals a   cos open parentheses n theta close parentheses and r equals a   sin open parentheses n theta close parentheses comma where n is a natural number.

  • When n is odd, the graph has n petals.
  • When n is even, the graph has 2 n petals.
Below are the graphs of selected polar equations. The same value of a is used in all graphs.

bold italic r bold equals bold italic a bold   bold sin open parentheses bold 3 bold theta close parentheses bold italic r bold equals bold italic a bold   bold sin open parentheses bold 4 bold theta close parentheses
A polar coordinate graph with an x-axis (polar axis) and a y-axis intersecting at the origin. The graph has concentric circles and radial lines spaced at regular angular intervals. A rose curve with three petals is arranged symmetrically around the origin, one petal on the negative y-axis, one to the right of the positive y-axis, and one to the left, equally spaced apart. A polar coordinate graph with an x-axis (polar axis) and a y-axis intersecting at the origin. The graph has concentric circles and radial lines spaced at regular angular intervals. A rose curve with eight petals is arranged symmetrically around the origin, with four petals extending to the right of the y-axis and four petals to the left.
bold italic r bold equals bold italic a bold   bold cos open parentheses bold 2 bold theta close parentheses bold italic r bold equals bold italic a bold   bold cos open parentheses bold 5 bold theta close parentheses
A polar coordinate graph with an x-axis (polar axis) and a y-axis intersecting at the origin. The graph has concentric circles and radial lines spaced at regular angular intervals. A rose curve with four petals is arranged symmetrically around the origin, with petals extending along the vertical and horizontal axes. A graph with a polar axis, represented by an x-axis, and the line theta equals pi divided by 2, intersecting at the origin. The graph has a five-petaled rose curve. The petals are arranged symmetrically around the center, with one petal symmetric with respect to the x-axis. the graph represents the polar function r equals a cosine of (5 times theta).

The last curve we will discuss is the spiral of Archimedes, which has the equation r equals k theta comma where k is a positive real number.

The graphs on two different intervals are shown below.

bold italic r bold equals bold italic theta bold comma bold space bold 0 bold less or equal than bold italic theta bold less or equal than bold 2 bold pi bold italic r bold equals bold italic theta bold comma bold space bold 0 bold less or equal than bold italic theta bold less or equal than bold 12 bold pi
The graph of a spiral curve. The curve starts at the origin, and opens outward as it moves counterclockwise, ending at the point that is 3 units to the right of the origin. The graph of a spiral curve. The curve starts at the origin, and opens outward as it moves counterclockwise, ending around six times, ending at a point to the right of the origin.

Notice that as theta increases, r increases, meaning that the curve steadily gets further away from the origin.

summary
In this lesson, you learned that one way to graph a polar equation is by using its corresponding rectangular form. When that is not convenient, you can test the polar equation for symmetry, find zeros and maximum values of open vertical bar bold r close vertical bar, then graph polar equations by plotting points to form the curve. There are also several forms of polar equations that are much easier to graph by referencing the graphs of classic polar equations.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.

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