1. Polar Coordinates Graphs of Polar Equations
Packet 2

2. Warm Ups – Day 1 1. Solve the equation algebraically:
2. Use synthetic division to prove that k is an upper bound:
k=5;
3. Find the domain and range for the function in #1.

3. Recall…
In the previous packet we learned how to graph polar equations of the form r = k and θ = k, where k is a constant.
In this section, we will learn how to graph more difficult types of polar equations.

4. Graphing Polar Equations
As with any type of equation, you can create a table of values for r and θ.
Then you can use the table to plot points in the polar plane.
Look at the examples in the packet.

5. Polar Graphs
You will notice that polar equations have graphs like the following:

6. Graphing Polar Equations on the TI-84
Hit the MODE key.
Arrow down to where it says Func (short for "function" which is a bit misleading since they are all functions).
Now, use the right arrow to choose Pol.
Hit ENTER. (*It's easy to forget this step, but it's crucial: until you hit ENTER you have not actually selected Pol, even though it looks like you have!)

7. Graphing Polar Equations on the TI-84
The calculator is now in polar coordinates mode. To see what that means, try this.
Hit the Y= key. Note that, instead of Y1=, Y2=, and so on, you now have r1= and so on.
In the r1= slot, type 5-5sin(θ)
Now hit the familiar X,T,θ,n key, and you get an unfamiliar result. In polar coordinates mode, this key gives you a θ instead of an X.
Finally, close off the parentheses and hit GRAPH.

8. Graphing Polar Equations on the TI-84
If you did everything right, you just asked the calculator to graph the polar equation r=5- 5sin(θ). The result looks a bit like a valentine.

9. Graphing Polar Equations on the TI-84
The WINDOW options are a little different in this mode too. You can still specify X and Y ranges, which define the viewing screen. But you can also specify the θ values that the calculator begins and ends with.

10. Graphing Polar Equations on the TI-84
For instance, you may limit the graph to 0<θ<π/2. This would not change the viewing window, but it would only draw part of the graph.

11. Graphing Polar Equations on the TI-84
Graph r = 3 sin 2θ
Enter the following window values:
Θmin = 0 Xmin = -6 Ymin = -4
θmax = 2π Xmax = 6 Ymax = 4
Θstep = π/24 Xscl = 1 Yscl = 1

12. Examples
Graph:
r = 2 cos θ
r = -2 cos θ
r = 1 – 2 cos θ

13. Examples
Graph:
r² = cos 2θ
Graph the system:
r = sin θ
r = 1