<< Chapter < Page Chapter >> Page >

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

Graphical

For the following exercises, test the equation for symmetry.

r = 3 3 cos θ

symmetric with respect to the polar axis

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

r = 5 sin 2 θ

symmetric with respect to the pole

Got questions? Get instant answers now!

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

r = 2 2 cos θ

cardioid
Graph of given cardioid.

Got questions? Get instant answers now!

r = 3 + 3 sin θ

cardioid
Graph of given cardioid.

Got questions? Get instant answers now!

r = 7 + 4 sin θ

one-loop/dimpled limaçon

Graph of given one-loop/dimpled limaçon
Got questions? Get instant answers now!

r = 5 + 4 cos θ

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

Got questions? Get instant answers now!

r = 1 + 3 sin θ

inner loop/two-loop limaçon

Graph of given inner loop/two-loop limaçon
Got questions? Get instant answers now!

r = 5 + 7 sin θ

inner loop/two-loop limaçon

Graph of given inner loop/two-loop limaçon
Got questions? Get instant answers now!

r = 5 + 6 cos θ

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

Got questions? Get instant answers now!

r 2 = 10 cos ( 2 θ )

lemniscate

Graph of given lemniscate (along horizontal axis)
Got questions? Get instant answers now!

r 2 = 10 sin ( 2 θ )

lemniscate

Graph of given lemniscate (along y=x)
Got questions? Get instant answers now!

r = 3 cos ( 2 θ )

rose curve

Graph of given rose curve - four petals.
Got questions? Get instant answers now!

r = 4 sin ( 4 θ )

rose curve

Graph of given rose curve - eight petals.
Got questions? Get instant answers now!

r = θ

Archimedes’ spiral

Graph of given Archimedes' spiral
Got questions? Get instant answers now!

r = 3 θ

Archimedes’ spiral

Graph of given Archimedes' spiral
Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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)
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

r = sin θ + θ , r = sin θ θ

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

r 1 = 6 4 cos θ , r 2 = 4

( 4 , π 3 ) , ( 4 , 5 π 3 )

Got questions? Get instant answers now!

r 1 = 1 + sin θ , r 2 = 3 sin θ

Got questions? Get instant answers now!

r 1 = 1 + cos θ , r 2 = 3 cos θ

( 3 2 , π 3 ) , ( 3 2 , 5 π 3 )

Got questions? Get instant answers now!

r 1 = cos ( 2 θ ) , r 2 = sin ( 2 θ )

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

r 1 = 3 , r 2 = 2 sin ( θ )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

r 1 = 1 + cos θ , r 2 = 1 sin θ

Got questions? Get instant answers now!
Practice Key Terms 9

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask