Gra ph the set of points whose polar coordinates satisfy the g iven equations and inequalities Relating Polar and Cartesian Coordinates Section 105b Relating Polar and Cartesian Coordinates Coordinate Conversion ID: 385467
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Slide1
Do Now - #18 on p.558
Gra
ph the set of points whose polar coordinates satisfy the
given equations and inequalities.Slide2
Relating Polar and Cartesian Coordinates
Section 10.5bSlide3
Relating Polar and Cartesian Coordinates
Coordinate Conversion
Equations:
Ray
Initial Ray
Common
originSlide4
Relating Polar and Cartesian Coordinates
Some curves are easier to work with in polar coordinates,others in Cartesian coordinates… Observe:
Polar Equation
Cartesian EquivalentSlide5
Relating Polar and Cartesian Coordinates
Find a polar equation for the circleSupport graphically.
Expand and simplify:
Conversion equations:
Algebra:
or
Check the graph!Slide6
Relating Polar and Cartesian Coordinates
Find a Cartesian equivalent for the polar equation. Identifythe graph.
(a)
Conversion equations
Completin
g the square
The graph of the equivalent Cartesian equation is a circle
w
ith radius 2 and center (2, 0).Slide7
Relating Polar and Cartesian Coordinates
Find a Cartesian equivalent for the polar equation. Identifythe graph.
(b)
Conversion equations
The graph of the equivalent Cartesian equation is a line
w
ith slope 2 and
y
-intercept –4.Slide8
Exploration 2
The polar curves and , wheren is an integer and , are
rose curves.
1. Graph for . Describe
t
he curves.
Graph window: [–4.7, 4.7] by [–3.1, 3.1]
The graphs are rose curves with 4 petals when ,
8 petals when , and 12 petals when .
2. What is the shortest length a -interval can have and still
p
roduce the graphs in (1)?
Shortest interval:Slide9
Exploration 2
The polar curves and , wheren is an integer and , are
rose curves.
3. Based on your observations in (1), describe the graph of
when
n
is a nonzero even integer.
The graph is a rose curve with petals.
4. Graph for . Describe
t
he curves.
Graph window: [–4.7, 4.7] by [–3.1, 3.1]
The graphs are rose curves with 3 petals when ,
5 petals when , and 7 petals when .Slide10
Exploration 2
The polar curves and , wheren is an integer and , are
rose curves.
5. What is the shortest length a -interval can have and still
p
roduce the graphs in (4)?
Shortest interval:
6. Based on your observations in (4), describe the graph of
when
n
is a nonzero odd integer different
from .
The graph is a rose curve with petals.Slide11
Guided Practice
Replace the polar equation by an
equivalent Cartesianequation. Then identify or describe the graph.
A parabola that opens to the rightSlide12
Guided Practice
Replace the polar equation by an
equivalent Cartesianequation. Then identify or describe the graph.
The union of
t
wo linesSlide13
Guided Practice
Replace the polar equation by an
equivalent Cartesianequation. Then identify or describe the graph.
A circle with center (0, 4)
a
nd radius 4Slide14
Guided Practice
Replace the Cartesian equation by an equivalent polar
equation. Support graphically.
How about the graph?Slide15
Guided Practice
Replace the Cartesian equation by an equivalent polar
equation. Support graphically.
How about the graph?
Graph: