Rewrite the polar equation
as a Cartesian equation.
The goal is to eliminate
and
and introduce
and
We clear the fraction, and then use substitution. In order to replace
with
and
we must use the expression
The Cartesian equation is
However, to graph it, especially using a graphing calculator or computer program, we want to isolate
When our entire equation has been changed from
and
to
and
we can stop, unless asked to solve for
or simplify. See
[link] .
The “hour-glass” shape of the graph is called a
hyperbola . Hyperbolas have many interesting geometric features and applications, which we will investigate further in
Analytic Geometry .
The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
To plot a point in the form
move in a counterclockwise direction from the polar axis by an angle of
and then extend a directed line segment from the pole the length of
in the direction of
If
is negative, move in a clockwise direction, and extend a directed line segment the length of
in the direction of
See
[link] .
If
is negative, extend the directed line segment in the opposite direction of
See
[link] .
To convert from polar coordinates to rectangular coordinates, use the formulas
and
See
[link] and
[link] .
To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:
and
See
[link] .
Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See
[link] ,
[link] , and
[link] .
Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See
[link] ,
[link] , and
[link] .
Section exercises
Verbal
How are polar coordinates different from rectangular coordinates?
For polar coordinates, the point in the plane depends on the angle from the positive
x- axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.
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?
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
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.
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
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
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?
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
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.