8.4 Polar coordinates: graphs  (Page 7/16)

 Page 7 / 16

Sketch the graph of $\text{\hspace{0.17em}}r=-\theta \text{\hspace{0.17em}}$ over the interval $\text{\hspace{0.17em}}\left[0,4\pi \right].$

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.

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 $\text{\hspace{0.17em}}\theta =\frac{\pi }{2},\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r.$
• The maximum value of a polar equation is found by substituting the value $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that leads to the maximum value of the trigonometric expression.
• The zeros of a polar equation are found by setting $\text{\hspace{0.17em}}r=0\text{\hspace{0.17em}}$ and solving for $\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ See [link] .
• Some formulas that produce the graph of a circle in polar coordinates are given by $\text{\hspace{0.17em}}r=a\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r=a\mathrm{sin}\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ See [link] .
• The formulas that produce the graphs of a cardioid are given by $\text{\hspace{0.17em}}r=a±b\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r=a±b\mathrm{sin}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}a>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}b>0,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\frac{a}{b}=1.\text{\hspace{0.17em}}$ See [link] .
• The formulas that produce the graphs of a one-loop limaçon are given by $\text{\hspace{0.17em}}r=a±b\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r=a±b\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}1<\frac{a}{b}<2.\text{\hspace{0.17em}}$ See [link] .
• The formulas that produce the graphs of an inner-loop limaçon are given by $\text{\hspace{0.17em}}r=a±b\mathrm{cos}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r=a±b\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}a>0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}b>0,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}a See [link] .
• The formulas that produce the graphs of a lemniscates are given by $\text{\hspace{0.17em}}{r}^{2}={a}^{2}\mathrm{cos}\text{\hspace{0.17em}}2\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{r}^{2}={a}^{2}\mathrm{sin}\text{\hspace{0.17em}}2\theta ,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\ne 0.$ See [link] .
• The formulas that produce the graphs of rose curves are given by $\text{\hspace{0.17em}}r=a\mathrm{cos}\text{\hspace{0.17em}}n\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r=a\mathrm{sin}\text{\hspace{0.17em}}n\theta ,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\ne 0;\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is even, there are $\text{\hspace{0.17em}}2n\text{\hspace{0.17em}}$ petals, and if $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is odd, there are $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ petals. See [link] and [link] .
• The formula that produces the graph of an Archimedes’ spiral is given by $\text{\hspace{0.17em}}r=\theta ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta \ge 0.\text{\hspace{0.17em}}$ See [link] .

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 $\text{\hspace{0.17em}}x$ -axis, symmetry with respect to the pole is similar to symmetry about the origin, and symmetric with respect to the line $\text{\hspace{0.17em}}\theta =\frac{\pi }{2}\text{\hspace{0.17em}}$ is similar to symmetry about the $\text{\hspace{0.17em}}y$ -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 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.

$r=5\mathrm{cos}\text{\hspace{0.17em}}3\theta$

$r=3-3\mathrm{cos}\text{\hspace{0.17em}}\theta$

symmetric with respect to the polar axis

$r=3+2\mathrm{sin}\text{\hspace{0.17em}}\theta$

$r=3\mathrm{sin}\text{\hspace{0.17em}}2\theta$

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

$r=4$

$r=2\theta$

no symmetry

$r=4\mathrm{cos}\text{\hspace{0.17em}}\frac{\theta }{2}$

$r=\frac{2}{\theta }$

no symmetry

$r=3\sqrt{1-{\mathrm{cos}}^{2}\theta }$

$r=\sqrt{5\mathrm{sin}\text{\hspace{0.17em}}2\theta }$

symmetric with respect to the pole

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

$r=3\mathrm{cos}\text{\hspace{0.17em}}\theta$

$r=4\mathrm{sin}\text{\hspace{0.17em}}\theta$

circle

$r=2+2\mathrm{cos}\text{\hspace{0.17em}}\theta$

$r=2-2\mathrm{cos}\text{\hspace{0.17em}}\theta$

cardioid

$r=5-5\mathrm{sin}\text{\hspace{0.17em}}\theta$

$r=3+3\mathrm{sin}\text{\hspace{0.17em}}\theta$

cardioid

$r=3+2\mathrm{sin}\text{\hspace{0.17em}}\theta$

$r=7+4\mathrm{sin}\text{\hspace{0.17em}}\theta$

one-loop/dimpled limaçon

$r=4+3\mathrm{cos}\text{\hspace{0.17em}}\theta$

$r=5+4\mathrm{cos}\text{\hspace{0.17em}}\theta$

one-loop/dimpled limaçon

$r=10+9\mathrm{cos}\text{\hspace{0.17em}}\theta$

$r=1+3\mathrm{sin}\text{\hspace{0.17em}}\theta$

inner loop/two-loop limaçon

$r=2+5\mathrm{sin}\text{\hspace{0.17em}}\theta$

$r=5+7\mathrm{sin}\text{\hspace{0.17em}}\theta$

inner loop/two-loop limaçon

$r=2+4\mathrm{cos}\text{\hspace{0.17em}}\theta$

$r=5+6\mathrm{cos}\text{\hspace{0.17em}}\theta$

inner loop/two-loop limaçon

${r}^{2}=36\mathrm{cos}\left(2\theta \right)$

${r}^{2}=10\mathrm{cos}\left(2\theta \right)$

lemniscate

${r}^{2}=4\mathrm{sin}\left(2\theta \right)$

${r}^{2}=10\mathrm{sin}\left(2\theta \right)$

lemniscate

$r=3\text{sin}\left(2\theta \right)$

$r=3\text{cos}\left(2\theta \right)$

rose curve

$r=5\text{sin}\left(3\theta \right)$

$r=4\text{sin}\left(4\theta \right)$

rose curve

$r=4\text{sin}\left(5\theta \right)$

$r=-\theta$

Archimedes’ spiral

$r=2\theta$

$r=-3\theta$

Archimedes’ spiral

Technology

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

$r=\frac{1}{\theta }$

$r=\frac{1}{\sqrt{\theta }}$

$r=2\mathrm{sin}\text{\hspace{0.17em}}\theta \mathrm{tan}\text{\hspace{0.17em}}\theta ,$ a cissoid

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

$r=5+\mathrm{cos}\left(4\theta \right)$

$r=2-\mathrm{sin}\left(2\theta \right)$

$r={\theta }^{2}$

$r=\theta +1$

$r=\theta \mathrm{sin}\text{\hspace{0.17em}}\theta$

$r=\theta \mathrm{cos}\text{\hspace{0.17em}}\theta$

For the following exercises, use a graphing utility to graph each pair of polar equations on a domain of $\text{\hspace{0.17em}}\left[0,4\pi \right]\text{\hspace{0.17em}}$ and then explain the differences shown in the graphs.

$r=\theta ,r=-\theta$

$r=\theta ,r=\theta +\mathrm{sin}\text{\hspace{0.17em}}\theta$

They are both spirals, but not quite the same.

$r=\mathrm{sin}\text{\hspace{0.17em}}\theta +\theta ,r=\mathrm{sin}\text{\hspace{0.17em}}\theta -\theta$

$r=2\mathrm{sin}\left(\frac{\theta }{2}\right),r=\theta \mathrm{sin}\left(\frac{\theta }{2}\right)$

Both graphs are curves with 2 loops. The equation with a coefficient of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ has two loops on the left, the equation with a coefficient of 2 has two loops side by side. Graph these from 0 to $\text{\hspace{0.17em}}4\pi \text{\hspace{0.17em}}$ to get a better picture.

$r=\mathrm{sin}\left(\mathrm{cos}\left(3\theta \right)\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}r=\mathrm{sin}\left(3\theta \right)$

On a graphing utility, graph $\text{\hspace{0.17em}}r=\mathrm{sin}\left(\frac{16}{5}\theta \right)\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[0,4\pi \right],\left[0,8\pi \right],\left[0,12\pi \right],\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left[0,16\pi \right].\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}r=\mathrm{sin}\text{\hspace{0.17em}}\theta +{\left(\mathrm{sin}\left(\frac{5}{2}\theta \right)\right)}^{3}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[0,4\pi \right].$

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

$\begin{array}{l}\begin{array}{l}\\ {r}_{1}=3\mathrm{sin}\left(3\theta \right)\end{array}\hfill \\ {r}_{2}=2\mathrm{sin}\left(3\theta \right)\hfill \\ {r}_{3}=\mathrm{sin}\left(3\theta \right)\hfill \end{array}$

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.

$\begin{array}{l}\begin{array}{l}\\ {r}_{1}=3+3\mathrm{cos}\text{\hspace{0.17em}}\theta \end{array}\hfill \\ {r}_{2}=2+2\mathrm{cos}\text{\hspace{0.17em}}\theta \hfill \\ {r}_{3}=1+\mathrm{cos}\text{\hspace{0.17em}}\theta \hfill \end{array}$

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

$\begin{array}{l}\begin{array}{l}\\ {r}_{1}=3\theta \end{array}\hfill \\ {r}_{2}=2\theta \hfill \\ {r}_{3}=\theta \hfill \end{array}$

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.

${r}_{1}=3+2\mathrm{sin}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}{r}_{2}=2$

${r}_{1}=6-4\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}{r}_{2}=4$

$\left(4,\frac{\pi }{3}\right),\left(4,\frac{5\pi }{3}\right)$

${r}_{1}=1+\mathrm{sin}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}{r}_{2}=3\mathrm{sin}\text{\hspace{0.17em}}\theta$

${r}_{1}=1+\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}{r}_{2}=3\mathrm{cos}\text{\hspace{0.17em}}\theta$

$\left(\frac{3}{2},\frac{\pi }{3}\right),\left(\frac{3}{2},\frac{5\pi }{3}\right)$

${r}_{1}=\mathrm{cos}\left(2\theta \right),\text{\hspace{0.17em}}{r}_{2}=\mathrm{sin}\left(2\theta \right)$

${r}_{1}={\mathrm{sin}}^{2}\left(2\theta \right),\text{\hspace{0.17em}}{r}_{2}=1-\mathrm{cos}\left(4\theta \right)$

$\left(0,\frac{\pi }{2}\right),\text{\hspace{0.17em}}\left(0,\pi \right),\text{\hspace{0.17em}}\left(0,\frac{3\pi }{2}\right),\text{\hspace{0.17em}}\left(0,2\pi \right)$

${r}_{1}=\sqrt{3},\text{\hspace{0.17em}}{r}_{2}=2\mathrm{sin}\left(\theta \right)$

${r}_{1}{}^{2}=\mathrm{sin}\text{\hspace{0.17em}}\theta ,{r}_{2}{}^{2}=\mathrm{cos}\text{\hspace{0.17em}}\theta$

$\left(\frac{\sqrt[4]{8}}{2},\frac{\pi }{4}\right),\text{\hspace{0.17em}}\left(\frac{\sqrt[4]{8}}{2},\frac{5\pi }{4}\right)\text{\hspace{0.17em}}$ and at $\text{\hspace{0.17em}}\theta =\frac{3\pi }{4},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{7\pi }{4}\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ since $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ is squared

${r}_{1}=1+\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}{r}_{2}=1-\mathrm{sin}\text{\hspace{0.17em}}\theta$

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
How can you tell what type of parent function a graph is ?
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)=
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
Darius
Thanks.
Thomas
Â
Thomas
It is the Â 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?
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
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
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
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.
how do you find the period of a sine graph
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
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
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert