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**Unformatted text preview: **e origin. Not only do we reach the origin when θ = 23 , a
π
theorem from Calculus5 states that the curve hugs the line θ = 23 as it approaches the origin.
5 The ‘tangents at the pole’ theorem from second semester Calculus. 11.5 Graphs of Polar Equations 803 r y 6
θ= 2π
3 4 2 x
2π
3 4π
3 π π
2 3π
2 2π θ −2 π
On the interval 23 , π , r ranges from 0 to −2. Since r ≤ 0, the curve passes through the
π
origin in the xy -plane, following the line θ = 23 and continues upwards through Quadrant IV
6 Since |r | is increasing from 0 to 2, the curve pulls away from
towards the positive x-axis.
the origin to ﬁnish at a point on the positive x-axis. r y 6 θ= 2π
3 4 2 x
2π
3
π
2 4π
3 π 3π
2 2π θ −2 π
Next, as θ progresses from π to 43 , r ranges from −2 to 0. Since r ≤ 0, we continue our
π
graph in the ﬁrst quadrant, heading into the origin along the line θ = 43 . 6
Recall that one way to visualize plotting polar coordinates (r, θ) with r < 0 is to start the rotation from the left
π
side of the pole - in this case, the negative x-axis. Rotating between 23 and π radians from the negative x-axis in
2π
this case determines the r...

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