College Trigonometry, 2011a

11.5 Graphs of Polar Equations 943
Next, as θ runs from π 2
to π, we see that r increases from 2 to 4. Picking up where we left
off, we gradually pull the graph away from the origin until we reach the negative x-axis.
6
r
θ runs from π 2 to π
y
4
x
2
π
2
π
3π
2
2π
θ
Over the interval [ π, 3π ]
2 , we see that r increases from 4 to 6. On the xy-plane, the curve
sweeps out away from the origin as it travels from the negative x-axis to the negative y-axis.
r
y
6
4
x
2
π
2
π
3π
2
2π
θ
θ runs from π to 3π 2
Finally, as θ takes on values from 3π 2
to 2π, r decreases from 6 back to 4. The graph on the
xy-plane pulls in from the negative y-axis to finish where we started.
r
y
6
4
x
2
π
2
π
3π
2
2π
θ
θ runs from 3π 2
to 2π
We leave it to the reader to verify that plotting points corresponding to values of θ outside
the interval [0, 2π] results in retracing portions of the curve, so we are finished.