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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] .

3 graphs side by side. (A) shows a ray extending into Q 1 and its symmetric version in Q 2. (B) shows a ray extending into Q 1 and its symmetric version in Q 4. (C) shows a ray extending into Q 1 and its symmetric version in Q 3. See caption for more information.
(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   ( r , θ ) :   ( − r , − θ )   for   θ = π 2   symmetry;   ( r , − θ )   for polar axis symmetry; and   ( − r , θ )   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   r = 2 sin   θ   for symmetry.

Test for each of the three types of symmetry.

1) Replacing   ( r , θ )   with   ( − r , − θ )   yields the same result. Thus, the graph is symmetric with respect to the line   θ = π 2 . − r = 2 sin ( − θ ) − r = −2 sin   θ Even-odd identity         r = 2 sin   θ Multiply   by   −1 Passed
2) Replacing   θ   with   − θ   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. r = 2 sin ( − θ ) r = −2 sin   θ Even-odd identity r = −2 sin   θ ≠ 2 sin   θ Failed
3) Replacing   r   with – r   changes the equation and fails the test. The graph may or may not be symmetric with respect to the pole. − r = 2 sin   θ    r = −2 sin   θ ≠ 2 sin   θ Failed
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Test the equation for symmetry:   r = − 2 cos   θ .

The equation fails the symmetry test with respect to the line   θ = π 2   and with respect to the pole. It passes the polar axis symmetry test.

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Graphing polar equations by plotting points

To graph in the rectangular coordinate system we construct a table of   x   and   y   values. To graph in the polar coordinate system we construct a table of   θ   and   r   values. We enter values of   θ   into a polar equation    and calculate   r .   However, using the properties of symmetry and finding key values of   θ   and   r   means fewer calculations will be needed.

Finding zeros and maxima

To find the zeros of a polar equation, we solve for the values of   θ   that result in   r = 0.   Recall that, to find the zeros of polynomial functions, we set the equation equal to zero and then solve for   x .   We use the same process for polar equations. Set   r = 0 ,   and solve for   θ .

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

Similarly, the maximum value of the sine function is 1 when   θ = π 2 ,   and if our polar equation is   r = 5 sin   θ ,   the value   θ = π 2   will yield the maximum   | r | .   We may find additional information by calculating values of   r   when   θ = 0.   These points would be polar axis intercepts, which may be helpful in drawing the graph and identifying the curve of a polar equation.

Questions & Answers

for the "hiking" mix, there are 1,000 pieces in the mix, containing 390.8 g of fat, and 165 g of protein. if there is the same amount of almonds as cashews, how many of each item is in the trail mix?
ADNAN Reply
linear speed of an object
Melissa Reply
an object is traveling around a circle with a radius of 13 meters .if in 20 seconds a central angle of 1/7 Radian is swept out what are the linear and angular speed of the object
Melissa
test
Matrix
how to find domain
Mohamed Reply
like this: (2)/(2-x) the aim is to see what will not be compatible with this rational expression. If x= 0 then the fraction is undefined since we cannot divide by zero. Therefore, the domain consist of all real numbers except 2.
Dan
define the term of domain
Moha
if a>0 then the graph is concave
Angel Reply
if a<0 then the graph is concave blank
Angel
what's a domain
Kamogelo Reply
The set of all values you can use as input into a function su h that the output each time will be defined, meaningful and real.
Spiro
how fast can i understand functions without much difficulty
Joe Reply
what is inequalities
Nathaniel
functions can be understood without a lot of difficulty. Observe the following: f(2) 2x - x 2(2)-2= 2 now observe this: (2,f(2)) ( 2, -2) 2(-x)+2 = -2 -4+2=-2
Dan
what is set?
Kelvin Reply
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
Divya Reply
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
158.5 This number can be developed by using algebra and logarithms. Begin by moving log(2) to the right hand side of the equation like this: t/100 log(2)= log(3) step 1: divide each side by log(2) t/100=1.58496250072 step 2: multiply each side by 100 to isolate t. t=158.49
Dan
what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
how to find x: 12x = 144 notice how 12 is being multiplied by x. Therefore division is needed to isolate x and whatever we do to one side of the equation we must do to the other. That develops this: x= 144/12 divide 144 by 12 to get x. addition: 12+x= 14 subtract 12 by each side. x =2
Dan
whats a domain
mike Reply
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.
Churlene Reply
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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