# 8.4 Polar coordinates: graphs  (Page 2/16)

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## Symmetry tests

A polar equation    describes a curve on the polar grid. The graph of a polar equation can be evaluated for three types of symmetry, as shown in [link] . (a) A graph is symmetric with respect to the line   θ = π 2   ( y -axis) if replacing   ( r , θ )   with   ( − r , − θ )   yields an equivalent equation. (b) A graph is symmetric with respect to the polar axis ( x -axis) if replacing   ( r , θ )   with   ( r , − θ )   or   ( − r , π− θ )   yields an equivalent equation. (c) A graph is symmetric with respect to the pole (origin) if replacing   ( r , θ )   with   ( − r , θ )   yields an equivalent equation.

Given a polar equation, test for symmetry.

1. Substitute the appropriate combination of components for $\text{\hspace{0.17em}}\left(r,\theta \right):$ $\text{\hspace{0.17em}}\left(-r,-\theta \right)\text{\hspace{0.17em}}$ for $\text{\hspace{0.17em}}\theta =\frac{\pi }{2}\text{\hspace{0.17em}}$ symmetry; $\text{\hspace{0.17em}}\left(r,-\theta \right)\text{\hspace{0.17em}}$ for polar axis symmetry; and $\text{\hspace{0.17em}}\left(-r,\theta \right)\text{\hspace{0.17em}}$ for symmetry with respect to the pole.
2. If the resulting equations are equivalent in one or more of the tests, the graph produces the expected symmetry.

## Testing a polar equation for symmetry

Test the equation $\text{\hspace{0.17em}}r=2\mathrm{sin}\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ for symmetry.

Test for each of the three types of symmetry.

 1) Replacing $\text{\hspace{0.17em}}\left(r,\theta \right)\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}\left(-r,-\theta \right)\text{\hspace{0.17em}}$ yields the same result. Thus, the graph is symmetric with respect to the line $\text{\hspace{0.17em}}\theta =\frac{\pi }{2}.$ 2) Replacing $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}-\theta \text{\hspace{0.17em}}$ does not yield the same equation. Therefore, the graph fails the test and may or may not be symmetric with respect to the polar axis. 3) Replacing $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ with $–r\text{\hspace{0.17em}}$ changes the equation and fails the test. The graph may or may not be symmetric with respect to the pole.

Test the equation for symmetry: $\text{\hspace{0.17em}}r=-2\mathrm{cos}\text{\hspace{0.17em}}\theta .$

The equation fails the symmetry test with respect to the line $\text{\hspace{0.17em}}\theta =\frac{\pi }{2}\text{\hspace{0.17em}}$ and with respect to the pole. It passes the polar axis symmetry test.

## Graphing polar equations by plotting points

To graph in the rectangular coordinate system we construct a table of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ values. To graph in the polar coordinate system we construct a table of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ values. We enter values of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ into a polar equation    and calculate $\text{\hspace{0.17em}}r.\text{\hspace{0.17em}}$ However, using the properties of symmetry and finding key values of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ means fewer calculations will be needed.

## Finding zeros and maxima

To find the zeros of a polar equation, we solve for the values of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that result in $\text{\hspace{0.17em}}r=0.\text{\hspace{0.17em}}$ Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ We use the same process for polar equations. Set $\text{\hspace{0.17em}}r=0,\text{\hspace{0.17em}}$ and solve for $\text{\hspace{0.17em}}\theta .$

For many of the forms we will encounter, the maximum value of a polar equation is found by substituting those values of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ into the equation that result in the maximum value of the trigonometric functions. Consider $\text{\hspace{0.17em}}r=5\mathrm{cos}\text{\hspace{0.17em}}\theta ;\text{\hspace{0.17em}}$ the maximum distance between the curve and the pole is 5 units. The maximum value of the cosine function is 1 when $\text{\hspace{0.17em}}\theta =0,\text{\hspace{0.17em}}$ so our polar equation is $\text{\hspace{0.17em}}5\mathrm{cos}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ and the value $\text{\hspace{0.17em}}\theta =0\text{\hspace{0.17em}}$ will yield the maximum $\text{\hspace{0.17em}}|r|.$

Similarly, the maximum value of the sine function is 1 when $\text{\hspace{0.17em}}\theta =\frac{\pi }{2},\text{\hspace{0.17em}}$ and if our polar equation is $\text{\hspace{0.17em}}r=5\mathrm{sin}\text{\hspace{0.17em}}\theta ,\text{\hspace{0.17em}}$ the value $\text{\hspace{0.17em}}\theta =\frac{\pi }{2}\text{\hspace{0.17em}}$ will yield the maximum $\text{\hspace{0.17em}}|r|.\text{\hspace{0.17em}}$ We may find additional information by calculating values of $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}\theta =0.\text{\hspace{0.17em}}$ These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich

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