It's time to get friendly with our graphing calculators. First, we need to understand how to graph polar functions by hand.
This is one of the many instances in calculus where it's helpful to use a calculator as a tool, but it's important to know what it's output means.
When checking the graph on the graphing calculator it can be helpful to spot-check points, especially at the boundaries of θ and simple angles like 0, 
, and 
to make sure r has the right values. Then look at what r is doing between those points to see if it makes sense.
Calculator Tip: If the calculator graph looks like jagged lines instead of looking curvy, try making the θ step size smaller (this may be called Δ θ on the calculator). This affects how carefully the calculator draws the graph.
Example 1
|
,
, the value of r = cos θ decreases from 1 to 0.
to π, the value r = cos θ goes from 0 to -1.
Example 2
Determine whether the following is a correct graph of
r = cos θ
for
|
.
we should have
and we do:
and 0 the value of r should be positive, and it is:Example 3
Determine whether the following is a correct graph of
r = sin θ
for
|
. 
Exercise 1
Without using a calculator, graph the function r = sin θ for
- 0 ≤ θ ≤ π
,
, the value r = sin θ increases from 0 to 1.
Exercise 2
Does the following graph match the given equation?
- r = 1 + cos θ for 0 ≤ θ ≤ π
Exercise 3
Determine if each graph is a reasonable graph of the given equation.
for 
.
These points are on the graph. When θ is in between 0 and π, the value of r should be positive. This makes sense with what we see on the graph,
so this is a reasonable graph.Exercise 4
Is the graph a reasonable graph of the given equation?
r = 2sin θ for 
we find that r = 2. Exercise 5
Is the graph a reasonable graph of the given equation?
r = 2sin θ for
