1. Do Now - #18 on p.558
Graph the set of points whose polar coordinates satisfy the
given equations and inequalities.

2. Relating Polar and Cartesian Coordinates
Section 10.5b

3. Relating Polar and Cartesian Coordinates
Coordinate Conversion
Equations:
Ray
Common
origin
Initial Ray

4. Relating Polar and Cartesian Coordinates
Some curves are easier to work with in polar coordinates,
others in Cartesian coordinates… Observe:
Polar Equation
Cartesian Equivalent

5. Relating Polar and Cartesian Coordinates
Find a polar equation for the circle
Support graphically.
Expand and simplify:
Conversion equations:
Algebra: or
Check the graph!

6. Relating Polar and Cartesian Coordinates
Find a Cartesian equivalent for the polar equation. Identify
the graph.
(a)
Conversion equations
Completing the square
The graph of the equivalent Cartesian equation is a circle
with radius 2 and center (2, 0).

7. Relating Polar and Cartesian Coordinates
Find a Cartesian equivalent for the polar equation. Identify
the graph.
(b)
Conversion equations
The graph of the equivalent Cartesian equation is a line
with slope 2 and y-intercept –4.

8. Exploration 2 The polar curves and , where
n is an integer and , are rose curves.
1. Graph for Describe
the 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
produce the graphs in (1)?
Shortest interval:

9. Exploration 2 The polar curves and , where
n 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
the 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

10. Exploration 2 The polar curves and , where
n is an integer and , are rose curves.
5. What is the shortest length a -interval can have and still
produce 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.

11. Guided Practice Replace the polar equation by an equivalent Cartesian
equation. Then identify or describe the graph.
A parabola that opens to the right

12. Guided Practice Replace the polar equation by an equivalent Cartesian
equation. Then identify or describe the graph.
The union of
two lines

13. Guided Practice Replace the polar equation by an equivalent Cartesian
equation. Then identify or describe the graph.
A circle with center (0, 4)
and radius 4

14. Guided Practice Replace the Cartesian equation by an equivalent polar
equation. Support graphically.
How about the graph?

15. Guided Practice Replace the Cartesian equation by an equivalent polar
equation. Support graphically.
Graph:
How about the graph?