4.4 Polar coordinates: graphs (Page 7/16)

Page 7 / 16

Try It

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

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.

Media

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 $θ = \frac{π}{2},$ 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 $r = a \pm b \cos θ$ and $r = a \pm b \sin θ,$ for $a > 0, b > 0,$ and $\frac{a}{b} = 1.$ See [link] .
The formulas that produce the graphs of a one-loop limaçon are given by $r = a \pm b \cos θ$ and $r = a \pm b \sin θ$ for $1 < \frac{a}{b} < 2.$ See [link] .
The formulas that produce the graphs of an inner-loop limaçon are given by $r = a \pm b \cos θ$ and $r = a \pm b \sin θ$ 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 $a \neq 0.$ 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 $a \neq 0;$ 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 $r = θ, θ \geq 0.$ 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 $θ = \frac{π}{2}$ is similar to symmetry about the $y$ -axis.

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

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 $θ = 0, \frac{π}{2}, π and \frac{3 π}{2},$ and sketch the graph.

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

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.

Graphical

For the following exercises, test the equation for symmetry.

$r = 5 \cos 3 θ$

$r = 3 - 3 \cos θ$

symmetric with respect to the polar axis

$r = 3 + 2 \sin θ$

$r = 3 \sin 2 θ$

symmetric with respect to the polar axis, symmetric with respect to the line $θ = \frac{π}{2},$ symmetric with respect to the pole

$r = 4$

$r = 2 θ$

no symmetry

$r = 4 \cos \frac{θ}{2}$

$r = \frac{2}{θ}$

no symmetry

$r = 3 \sqrt{1 - \cos^{2} θ}$

$r = \sqrt{5 \sin 2 θ}$

symmetric with respect to the pole

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

$r = 3 \cos θ$

$r = 4 \sin θ$

circle
Graph of given circle.

$r = 2 + 2 \cos θ$

$r = 2 - 2 \cos θ$

cardioid
Graph of given cardioid.

$r = 5 - 5 \sin θ$

$r = 3 + 3 \sin θ$

cardioid
Graph of given cardioid.

$r = 3 + 2 \sin θ$

$r = 7 + 4 \sin θ$

one-loop/dimpled limaçon

$r = 4 + 3 \cos θ$

$r = 5 + 4 \cos θ$

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

$r = 10 + 9 \cos θ$

$r = 1 + 3 \sin θ$

inner loop/two-loop limaçon

$r = 2 + 5 \sin θ$

$r = 5 + 7 \sin θ$

inner loop/two-loop limaçon

$r = 2 + 4 \cos θ$

$r = 5 + 6 \cos θ$

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

$r^{2} = 36 \cos (2 θ)$

$r^{2} = 10 \cos (2 θ)$

lemniscate

$r^{2} = 4 \sin (2 θ)$

$r^{2} = 10 \sin (2 θ)$

lemniscate

$r = 3 sin (2 θ)$

$r = 3 cos (2 θ)$

rose curve

$r = 5 sin (3 θ)$

$r = 4 sin (4 θ)$

rose curve

$r = 4 sin (5 θ)$

$r = - θ$

Archimedes’ spiral

$r = 2 θ$

$r = - 3 θ$

Archimedes’ spiral

Technology

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

$r = \frac{1}{θ}$

$r = \frac{1}{\sqrt{θ}}$

$r = 2 \sin θ \tan θ,$ a cissoid

$r = 2 \sqrt{1 - \sin^{2} θ}$ , a hippopede

Graph of given hippopede (two circles that are centered along the x-axis and meet at the origin)

$r = 5 + \cos (4 θ)$

$r = 2 - \sin (2 θ)$

$r = θ^{2}$

$r = θ + 1$

Graph of given equation. Similar to original Archimedes' spiral.

$r = θ \sin θ$

$r = θ \cos θ$

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.

$r = θ, r = - θ$

$r = θ, r = θ + \sin θ$

They are both spirals, but not quite the same.

$r = \sin θ + θ, r = \sin θ - θ$

$r = 2 \sin (\frac{θ}{2}), r = θ \sin (\frac{θ}{2})$

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.

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

On a graphing utility, graph $r = \sin (\frac{16}{5} θ)$ 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.

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

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

\begin{array}{l} \begin{array}{l} r_{1} = 3 \sin (3 θ) \end{array} \\ r_{2} = 2 \sin (3 θ) \\ r_{3} = \sin (3 θ) \end{array}

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

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

\begin{array}{l} \begin{array}{l} r_{1} = 3 + 3 \cos θ \end{array} \\ r_{2} = 2 + 2 \cos θ \\ r_{3} = 1 + \cos θ \end{array}

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

\begin{array}{l} \begin{array}{l} r_{1} = 3 θ \end{array} \\ r_{2} = 2 θ \\ r_{3} = θ \end{array}

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

Extensions

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

$r_{1} = 3 + 2 \sin θ, r_{2} = 2$

$r_{1} = 6 - 4 \cos θ, r_{2} = 4$

$(4, \frac{π}{3}), (4, \frac{5 π}{3})$

$r_{1} = 1 + \sin θ, r_{2} = 3 \sin θ$

$r_{1} = 1 + \cos θ, r_{2} = 3 \cos θ$

$(\frac{3}{2}, \frac{π}{3}), (\frac{3}{2}, \frac{5 π}{3})$

$r_{1} = \cos (2 θ), r_{2} = \sin (2 θ)$

$r_{1} = \sin^{2} (2 θ), r_{2} = 1 - \cos (4 θ)$

$(0, \frac{π}{2}), (0, π), (0, \frac{3 π}{2}), (0, 2 π)$

$r_{1} = \sqrt{3}, r_{2} = 2 \sin (θ)$

$r_{1}^{2} = \sin θ, r_{2}^{2} = \cos θ$

$(\frac{\sqrt[4]{8}}{2}, \frac{π}{4}), (\frac{\sqrt[4]{8}}{2}, \frac{5 π}{4})$ and at $θ = \frac{3 π}{4}, \frac{7 π}{4}$ since $r$ is squared

$r_{1} = 1 + \cos θ, r_{2} = 1 - \sin θ$

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Source: OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2

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