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Sketch the graph of r = θ over the interval [ 0 , 4 π ] .

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Summary of curves

We have explored a number of seemingly complex polar curves in this section. [link] and [link] summarize the graphs and equations for each of these curves.

Four graphs side by side - a summary. (A) is a circle: r=asin(theta) or r=acos(theta). (B) is a cardioid: r= a + or - bcos(theta), or r = a + or - b sin(theta). a>0, b>0, a/b=1. (C) is one-loop limaçons. r= a + or - bcos(theta), or r= a + or - bsin(theta). a>0, b>0, 1<a/b<2. (D) is inner-loop limaçons. R = a + or - bcos(theta), or r = a + or - bsin(theta). A>0, b>0, a<b.
Four graphs side by side - a summary. (A) is lemniscates. R^2 = a^2cos(2theta), or r^2=a^2sin(2theta). a is not equal to 0. (B) is a rsose curve (n even). R = acos(ntheta), or r=asin(ntheta). N is even, and there are 2n petals. (C) is a rose curve (n odd). R = acos(ntheta), or r=asin(theta). N is odd, and there are n petals. (D) is an Archimedes's spiral. R=theta, and theta >=0.

Access these online resources for additional instruction and practice with graphs of polar coordinates.

Key concepts

  • It is easier to graph polar equations if we can test the equations for symmetry with respect to the line θ = π 2 , the polar axis, or the pole.
  • There are three symmetry tests that indicate whether the graph of a polar equation will exhibit symmetry. If an equation fails a symmetry test, the graph may or may not exhibit symmetry. See [link] .
  • Polar equations may be graphed by making a table of values for θ and r .
  • The maximum value of a polar equation is found by substituting the value θ that leads to the maximum value of the trigonometric expression.
  • The zeros of a polar equation are found by setting r = 0 and solving for θ . See [link] .
  • Some formulas that produce the graph of a circle in polar coordinates are given by r = a cos θ and r = a sin θ . See [link] .
  • The formulas that produce the graphs of a cardioid are given by r = a ± b cos θ and r = a ± b sin θ , for a > 0 , b > 0 , and a b = 1. See [link] .
  • The formulas that produce the graphs of a one-loop limaçon are given by r = a ± b cos θ and r = a ± b sin θ for 1 < a b < 2. See [link] .
  • The formulas that produce the graphs of an inner-loop limaçon are given by r = a ± b cos θ and r = a ± b sin θ for a > 0 , b > 0 , and a < b . See [link] .
  • The formulas that produce the graphs of a lemniscates are given by r 2 = a 2 cos 2 θ and r 2 = a 2 sin 2 θ , where a 0. See [link] .
  • The formulas that produce the graphs of rose curves are given by r = a cos n θ and r = a sin n θ , where a 0 ; if n is even, there are 2 n petals, and if n is odd, there are n petals. See [link] and [link] .
  • The formula that produces the graph of an Archimedes’ spiral is given by r = θ , θ 0. See [link] .

Section exercises

Verbal

Describe the three types of symmetry in polar graphs, and compare them to the symmetry of the Cartesian plane.

Symmetry with respect to the polar axis is similar to symmetry about the x -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line θ = π 2 is similar to symmetry about the y -axis.

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Which of the three types of symmetries for polar graphs correspond to the symmetries with respect to the x -axis, y -axis, and origin?

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What are the steps to follow when graphing polar equations?

Test for symmetry; find zeros, intercepts, and maxima; make a table of values. Decide the general type of graph, cardioid, limaçon, lemniscate, etc., then plot points at θ = 0 , π 2 , π and  3 π 2 , and sketch the graph.

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Describe the shapes of the graphs of cardioids, limaçons, and lemniscates.

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What part of the equation determines the shape of the graph of a polar equation?

The shape of the polar graph is determined by whether or not it includes a sine, a cosine, and constants in the equation.

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Graphical

For the following exercises, test the equation for symmetry.

r = 3 3 cos θ

symmetric with respect to the polar axis

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r = 3 sin 2 θ

symmetric with respect to the polar axis, symmetric with respect to the line θ = π 2 , symmetric with respect to the pole

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r = 5 sin 2 θ

symmetric with respect to the pole

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For the following exercises, graph the polar equation. Identify the name of the shape.

r = 2 2 cos θ

cardioid
Graph of given cardioid.

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r = 3 + 3 sin θ

cardioid
Graph of given cardioid.

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r = 7 + 4 sin θ

one-loop/dimpled limaçon

Graph of given one-loop/dimpled limaçon
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r = 5 + 4 cos θ

one-loop/dimpled limaçon
Graph of given one-loop/dimpled limaçon

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r = 1 + 3 sin θ

inner loop/two-loop limaçon

Graph of given inner loop/two-loop limaçon
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r = 5 + 7 sin θ

inner loop/two-loop limaçon

Graph of given inner loop/two-loop limaçon
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r = 5 + 6 cos θ

inner loop/two-loop limaçon
Graph of given inner loop/two-loop limaçon

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r 2 = 10 cos ( 2 θ )

lemniscate

Graph of given lemniscate (along horizontal axis)
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r 2 = 10 sin ( 2 θ )

lemniscate

Graph of given lemniscate (along y=x)
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r = 3 cos ( 2 θ )

rose curve

Graph of given rose curve - four petals.
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r = 4 sin ( 4 θ )

rose curve

Graph of given rose curve - eight petals.
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r = θ

Archimedes’ spiral

Graph of given Archimedes' spiral
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r = 3 θ

Archimedes’ spiral

Graph of given Archimedes' spiral
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Technology

For the following exercises, use a graphing calculator to sketch the graph of the polar equation.

r = 2 sin θ tan θ , a cissoid

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r = 2 1 sin 2 θ , a hippopede

Graph of given hippopede (two circles that are centered along the x-axis and meet at the origin)
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For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of [ 0 , 4 π ] and then explain the differences shown in the graphs.

r = θ , r = θ + sin θ

They are both spirals, but not quite the same.

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r = sin θ + θ , r = sin θ θ

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r = 2 sin ( θ 2 ) , r = θ sin ( θ 2 )

Both graphs are curves with 2 loops. The equation with a coefficient of θ has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to 4 π to get a better picture.

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r = sin ( cos ( 3 θ ) ) r = sin ( 3 θ )

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On a graphing utility, graph r = sin ( 16 5 θ ) on [ 0 , 4 π ] , [ 0 , 8 π ] , [ 0 , 12 π ] , and [ 0 , 16 π ] . Describe the effect of increasing the width of the domain.

When the width of the domain is increased, more petals of the flower are visible.

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On a graphing utility, graph and sketch r = sin θ + ( sin ( 5 2 θ ) ) 3 on [ 0 , 4 π ] .

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On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

r 1 = 3 sin ( 3 θ ) r 2 = 2 sin ( 3 θ ) r 3 = sin ( 3 θ )

The graphs are three-petal, rose curves. The larger the coefficient, the greater the curve’s distance from the pole.

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On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

r 1 = 3 + 3 cos θ r 2 = 2 + 2 cos θ r 3 = 1 + cos θ
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On a graphing utility, graph each polar equation. Explain the similarities and differences you observe in the graphs.

r 1 = 3 θ r 2 = 2 θ r 3 = θ

The graphs are spirals. The smaller the coefficient, the tighter the spiral.

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Extensions

For the following exercises, draw each polar equation on the same set of polar axes, and find the points of intersection.

r 1 = 3 + 2 sin θ , r 2 = 2

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r 1 = 6 4 cos θ , r 2 = 4

( 4 , π 3 ) , ( 4 , 5 π 3 )

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r 1 = 1 + sin θ , r 2 = 3 sin θ

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r 1 = 1 + cos θ , r 2 = 3 cos θ

( 3 2 , π 3 ) , ( 3 2 , 5 π 3 )

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r 1 = cos ( 2 θ ) , r 2 = sin ( 2 θ )

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r 1 = sin 2 ( 2 θ ) , r 2 = 1 cos ( 4 θ )

( 0 , π 2 ) , ( 0 , π ) , ( 0 , 3 π 2 ) , ( 0 , 2 π )

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r 1 = 3 , r 2 = 2 sin ( θ )

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r 1 2 = sin θ , r 2 2 = cos θ

( 8 4 2 , π 4 ) , ( 8 4 2 , 5 π 4 ) and at θ = 3 π 4 , 7 π 4 since r is squared

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r 1 = 1 + cos θ , r 2 = 1 sin θ

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Questions & Answers

find the equation of the line if m=3, and b=-2
Ashley Reply
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
Fiston Reply
x=-b+_Гb2-(4ac) ______________ 2a
Ahlicia Reply
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
Carlos Reply
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
Brad
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
Mary Reply
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
Practice Key Terms 9

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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