# 8.4 Polar coordinates: graphs

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In this section you will:
• Test polar equations for symmetry.
• Graph polar equations by plotting points.

The planets move through space in elliptical, periodic orbits about the sun, as shown in [link] . They are in constant motion, so fixing an exact position of any planet is valid only for a moment. In other words, we can fix only a planet’s instantaneous position. This is one application of polar coordinates    , represented as $\text{\hspace{0.17em}}\left(r,\theta \right).\text{\hspace{0.17em}}$ We interpret $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ as the distance from the sun and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ as the planet’s angular bearing, or its direction from a fixed point on the sun. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates.

## Testing polar equations for symmetry

Just as a rectangular equation such as $\text{\hspace{0.17em}}y={x}^{2}\text{\hspace{0.17em}}$ describes the relationship between $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ on a Cartesian grid, a polar equation describes a relationship between $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ on a polar grid. Recall that the coordinate pair $\text{\hspace{0.17em}}\left(r,\theta \right)\text{\hspace{0.17em}}$ indicates that we move counterclockwise from the polar axis (positive x -axis) by an angle of $\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ and extend a ray from the pole (origin) $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ units in the direction of $\text{\hspace{0.17em}}\theta .\text{\hspace{0.17em}}$ All points that satisfy the polar equation are on the graph.

Symmetry is a property that helps us recognize and plot the graph of any equation. If an equation has a graph that is symmetric with respect to an axis, it means that if we folded the graph in half over that axis, the portion of the graph on one side would coincide with the portion on the other side. By performing three tests, we will see how to apply the properties of symmetry to polar equations. Further, we will use symmetry (in addition to plotting key points, zeros, and maximums of $\text{\hspace{0.17em}}r\right)\text{\hspace{0.17em}}$ to determine the graph of a polar equation.

In the first test, we consider symmetry with respect to the line $\text{\hspace{0.17em}}\theta =\frac{\pi }{2}\text{\hspace{0.17em}}$ ( y -axis). We replace $\text{\hspace{0.17em}}\left(r,\theta \right)\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}\left(-r,-\theta \right)\text{\hspace{0.17em}}$ to determine if the new equation is equivalent to the original equation. For example, suppose we are given the equation $\text{\hspace{0.17em}}r=2\mathrm{sin}\text{\hspace{0.17em}}\theta ;$

This equation exhibits symmetry with respect to the line $\text{\hspace{0.17em}}\theta =\frac{\pi }{2}.$

In the second test, we consider symmetry with respect to the polar axis ( $\text{\hspace{0.17em}}x$ -axis). We replace $\text{\hspace{0.17em}}\left(r,\theta \right)\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}\left(r,-\theta \right)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(-r,\pi -\theta \right)\text{\hspace{0.17em}}$ to determine equivalency between the tested equation and the original. For example, suppose we are given the equation $\text{\hspace{0.17em}}r=1-2\mathrm{cos}\text{\hspace{0.17em}}\theta .$

The graph of this equation exhibits symmetry with respect to the polar axis.

In the third test, we consider symmetry with respect to the pole (origin). We replace $\text{\hspace{0.17em}}\left(r,\theta \right)\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}\left(-r,\theta \right)\text{\hspace{0.17em}}$ to determine if the tested equation is equivalent to the original equation. For example, suppose we are given the equation $\text{\hspace{0.17em}}r=2\mathrm{sin}\left(3\theta \right).$

$\begin{array}{c}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}r=2\mathrm{sin}\left(3\theta \right)\\ -r=2\mathrm{sin}\left(3\theta \right)\end{array}$

The equation has failed the symmetry test , but that does not mean that it is not symmetric with respect to the pole. Passing one or more of the symmetry tests verifies that symmetry will be exhibited in a graph. However, failing the symmetry tests does not necessarily indicate that a graph will not be symmetric about the line $\text{\hspace{0.17em}}\theta =\frac{\pi }{2},\text{\hspace{0.17em}}$ the polar axis, or the pole. In these instances, we can confirm that symmetry exists by plotting reflecting points across the apparent axis of symmetry or the pole. Testing for symmetry is a technique that simplifies the graphing of polar equations, but its application is not perfect.

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
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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
can I see the picture
How would you find if a radical function is one to one?