Topic 3.14B

Polar Function Graphs

Part A covered circles and roses. Part B covers one more family, the limaçons, and gives you two easy tools that work on ANY polar graph: the four-point sketch and the rectangular graph of $r$ .

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01
$r = 0$ → the curve passes through the pole. $r < 0$ → the point is on the opposite side.

Practice 43 problems Next: Chapter 3.15 Rates of Change in Polar Functions

Limaçons

r=a+b\cos\theta\quad\text{or}\quad r=a+b\sin\theta

A limaçon (say "LEE-ma-son") is a constant plus a sinusoid. There are four shapes, and which one you get depends only on the SIZES of $a$ and $b$ (take $a > 0$ ):

The four kinds of limaçon.

CONCEPT

Classify in one step: compare a with $|b|$

$> a < |b|$ → inner loop ( $r$ becomes negative for a while).

$> a = |b|$ → cardioid, a heart shape ( $r$ just touches 0 once).

$|b| < a < 2|b|$ → dimpled (a dent, but no loop).

$a \ge 2|b|$ → convex (round, no dent).

Then find the direction: $\cos \theta$ versions point left/right, $\sin \theta$ versions point up/down. The widest part is where $r = a + |b|$ .

Example	Compare	Type	Largest r and where
r = 2 + 4 cos θ	2 < 4	inner loop	6, at θ = 0 (right)
r = 3 + 3 sin θ	3 = 3	cardioid	6, at θ = π/2 (up)
r = 4 + 3 cos θ	3 < 4 < 6	dimpled	7, at θ = 0 (right)
r = 5 − 2 sin θ	5 ≥ 4	convex	7, at θ = 3π/2 (down)

COMMON MISTAKE

Deciding the type from the SIGN of $b$ , so that $r = 5 - 2 \sin \theta$ is called something different from $r = 5 + 2 \sin \theta$ .

The type depends only on the sizes a and $|b|$ . The sign of $b$ only decides which way the curve points: $5 + 2 \sin \theta$ is widest upward, $5 - 2 \sin \theta$ is widest downward. Both are convex because $5 \ge 2 \cdot 2$ .

CONCEPT

Try it yourself

Classify each limaçon and say which way it is widest.

$r = 3 + 5 \sin \theta 2$ . $r = 4 - 4 \cos \theta 3$ . $r = 5 + 3 \sin \theta 4$ . $r = 7 + 2 \cos \theta$

Answers: 1. Inner loop ( $3 < 5$ ), widest up. 2. Cardioid ( $4 = 4$ ), widest LEFT, because $r = 8$ at $\theta = \pi$ . 3. Dimpled ( $3 < 5 < 6$ ), widest up. 4. Convex ( $7 \ge 4$ ), widest right.

Quick check

Classify $r = 2 + 3\cos\theta$ .

The Four-Point Sketch

You do not need a big table to sketch a limaçon. Four points — one in each direction — are enough to get the shape.

CONCEPT

Four-point sketch

Find $r$ at $\theta = 0$ , $\pi /2$ , $\pi$ , and $3\pi /2$ (right, up, left, down).
Plot them. If an $r$ is negative, the point goes on the OPPOSITE side.
If $r = 0$ somewhere, the curve passes through the pole there.
Connect smoothly, going around counterclockwise in the order of $\theta$ .

If you want more accuracy, use eight angles instead of four — every $\pi /4$ . Here is $r = 4 + 3 \cos \theta$ :

θ	0	π/4	π/2	3π/4	π	5π/4	3π/2	7π/4
r	7	6.12	4	1.88	1	1.88	4	6.12

Eight points, numbered in order of $\theta$ , then connected.

Reading the table tells you the shape before you draw anything. $r$ starts at its largest, 7, shrinks to its smallest, 1, at $\theta = \pi$ , and grows back. Because the smallest $r$ is 1 — small, but still positive — the curve squeezes in close to the pole on the left without touching it. That squeeze is the DIMPLE. If the smallest $r$ had been 0 you would get a cardioid, and if it had been negative you would get an inner loop.

Worked example

Example 1. Sketch $r = 3 - 3 \cos \theta$ and $r = 2 + 4 \cos \theta$ .

01
$r = 3 - 3 \cos \theta$ : $r(0) = 0$ , $r(\pi /2) = 3$ , $r(\pi ) = 6$ , $r(3\pi /2) = 3$ . It starts AT the pole, goes up to 3, left to 6, down to 3, and back. Since $a = |b| = 3$ , it is a cardioid, widest on the left.
02
$r = 2 + 4 \cos \theta$ : $r(0) = 6$ , $r(\pi /2) = 2$ , $r(\pi ) = -2$ , $r(3\pi /2) = 2$ . The value $r(\pi ) = -2$ is negative: facing left and walking backward lands at $(2, 0)$ , on the RIGHT, inside the big loop.

Where does $r = 2 + 4 \cos \theta$ pass through the pole? Solve $2 + 4 \cos \theta = 0$ : $\cos \theta = -1/2$ , so $\theta = 2\pi /3$ and $4\pi /3$ (3.10A). Between these angles $r < 0$ , and the curve draws the small inner loop.

The four key points (dark dots). Blue: $r > 0$ . Red: $r < 0$ — the inner loop.

The inner loop of $r = 2 + 4 \cos \theta$ deserves a slow walk-through, because it is where most mistakes happen. Follow the three stages below, one θ-interval at a time.

θ-interval	what r does	what the point does
0 → 2π/3	falls from 6 to 0	starts far right, swings counterclockwise, arrives at the pole
2π/3 → π	falls from 0 to −2	leaves the pole on the OPPOSITE side, moving right to (2, 0)
π → 4π/3	rises from −2 to 0	comes back to the pole (still on the opposite side)
4π/3 → 2π	rises from 0 to 6	draws the rest of the outer curve and closes it

Three stages. The middle picture is the whole inner loop: it is drawn entirely while $r$ is negative.

Notice that the inner loop is traced on the RIGHT even though those angles (between $2\pi /3$ and $4\pi /3$ ) all point to the left. That is the whole story of a negative $r$ .

Quick check

For $r = 1 + 2\sin\theta$ , at which $\theta$ in $[0, 2\pi)$ does the curve pass through the pole?

From the Rectangular Graph to the Polar Graph

KEY RULE

$r = 0$ → the curve passes through the pole. $r < 0$ → the point is on the opposite side.

Worked example

Example 2. Use the rectangular graph of $r = 1 + 2 \sin \theta$ to sketch its polar graph for $0 \le \theta \le 2\pi$ , and explain where the inner loop comes from.

01
Zeros: $1 + 2 \sin \theta = 0$ → $\sin \theta = -1/2$ → $\theta = 7\pi /6$ and $11\pi /6$ (3.10A, Example 1). The curve passes through the pole at these angles.
02
Split the θ-axis at the key angles. A (0 to $\pi /2$ ): $r$ grows from 1 to 3, moving away. B ( $\pi /2$ to $7\pi /6$ ): $r$ shrinks from 3 to 0, coming back to the pole.
03
$C$ ( $7\pi /6$ to $11\pi /6$ ): $r$ is NEGATIVE, lowest value −1 at $\theta = 3\pi /2$ . Facing down and walking backward 1 unit puts the point at $(0, 1)$ , ABOVE the pole. All of segment $C$ lands on the opposite side and forms the inner loop.
04
$D$ ( $11\pi /6$ to $2\pi$ ): $r$ grows from 0 back to 1, closing the curve at $(1, 0)$ . Type check: $a = 1 < |b| = 2$ , inner loop ✓.

Matching colors: the negative part $C$ of the rectangular graph becomes the inner loop of the polar graph.

COMMON MISTAKE

Reading the lowest point of the rectangular graph, $r = -1$ at $\theta = 3\pi /2$ , as "the curve is 1 unit below the pole."

The rectangular graph shows $r$ , not height. $r = -1$ at $\theta = 3\pi /2$ means face down and walk backward, which is 1 unit UP. The lowest point of the rectangular graph becomes the TOP of the inner loop.

Worked example

Example 3. Now flip the sign: sketch $r = 1 - 2 \sin \theta$ . Where is its inner loop?

01
Zeros: $1 - 2 \sin \theta = 0$ → $\sin \theta = 1/2$ → $\theta = \pi /6$ and $5\pi /6$ .
02
Between $\pi /6$ and $5\pi /6$ , $\sin \theta > 1/2$ , so $r < 0$ . The lowest value is $r(\pi /2) = 1 - 2 = -1$ : facing UP and walking backward puts the point at $(0, -1)$ , BELOW the pole.
03
So this time the inner loop hangs below the pole, and the big loop is widest downward: $r(3\pi /2) = 1 + 2 = 3$ .

$r = 1 - 2 \sin \theta$ : the red (negative) part becomes an inner loop below the pole.

CONCEPT

What the rectangular graph of $r$ tells you

$r > 0$ : the point is in the direction $\theta$ . $r < 0$ : it is in the direction $\theta + \pi$ .

$r = 0$ : the curve is at the pole.

$|r|$ large: the point is far from the pole.

$|r|$ increasing: moving away from the pole; $|r|$ decreasing: moving toward it. (3.15 measures how fast.)

Common slips

Deciding the type from the sign of $b$ , so that $r = 5 - 2 \sin \theta$ is called something different from $r = 5 + 2 \sin \theta$ .

The type depends only on the sizes a and $|b|$ . The sign of $b$ only decides which way the curve points: $5 + 2 \sin \theta$ is widest upward, $5 - 2 \sin \theta$ is widest downward. Both are convex because $5 \ge 2 \cdot 2$ .
Reading the lowest point of the rectangular graph, $r = -1$ at $\theta = 3\pi /2$ , as "the curve is 1 unit below the pole."

The rectangular graph shows $r$ , not height. $r = -1$ at $\theta = 3\pi /2$ means face down and walk backward, which is 1 unit up. The lowest point of the rectangular graph becomes the top of the inner loop.

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13 cards · Polar, Which polar curve?

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Recap card

4 lines to re-read the night before.

01
Limaçons $r = a + b \cos \theta$ or $a + b \sin \theta$ : inner loop if $a < |b|$ , cardioid if $a = |b|$ , dimpled if $|b| < a < 2|b|$ , convex if $a \ge 2|b|$ . The sign of $b$ only changes the direction.
02
Four-point sketch: find $r$ at 0, $\pi /2$ , $\pi$ , $3\pi /2$ , plot (negative $r$ on the opposite side), and connect in order.
03
Zeros of $r$ are where the curve passes through the pole; stretches with $r < 0$ land on the opposite side — that is where inner loops come from.
04
Coming up in 3.15: how fast a point moves toward or away from the pole.

Practice 43 problems Next: Chapter 3.15 Rates of Change in Polar Functions

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Polar Function Graphs

Limaçons

The Four-Point Sketch

From the Rectangular Graph to the Polar Graph

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