How many petals does r = 2cos(3θ) have? [1 point]

College Board AP 2024 (style)-style practice: Section II (free response, calculator allowed). Consider r = 2 + 2sinθ. (a) Find r at θ = 0, (π)/(2), π, (3π)/(2). (b) Identify the type of curve and state where it passes through the pole.

A 4-point question on reading a polar graph. (a) Values (2 points): r(0) = 2 + 2sin 0 = 2; r((π)/(2)) = 2 + 2(1) = 4; r(π) = 2 + 2(0) = 2; r((3π)/(2)) = 2 + 2(-1) = 0. (b) Curve and pole (2 points): since the constant and the coefficient of sinθ are equal (2 = 2), the curve is a cardioid (a limacon with an inner loop reduced to a cusp). It passes through the pole where r = 0, at θ = (3π)/(2).

United StatesPrecalculusSyllabus dot point

How do you graph a polar function r = f(theta), and what shapes do the standard polar functions produce?

Topic 3.14 Polar Function Graphs: construct and interpret the graph of a polar function r = f(theta), including circles, roses, limacons and spirals.

A focused answer to AP Precalculus Topic 3.14, covering how to graph a polar function r = f(theta) by plotting radius against angle, the standard polar shapes (circles, roses, limacons, spirals), and how the sign of r affects the graph.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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Quick answer

A polar function $r = f(\theta)$ gives the distance from the pole as a function of the angle. To graph it, evaluate $r$ at a sequence of angles and plot each point $(r, \theta)$ , joining them as $\theta$ increases. Several standard forms recur: $r = a$ is a circle of radius $a$ centered at the pole; $r = a\cos\theta$ or $r = a\sin\theta$ is a circle through the pole; $r = a + b\sin\theta$ or $a + b\cos\theta$ is a limacon (a cardioid when $a = b$ ); $r = a\cos(n\theta)$ or $a\sin(n\theta)$ is a rose with $n$ or $2n$ petals; and $r = a\theta$ is a spiral. The curve passes through the pole wherever $r = 0$ . When $r$ is negative, the point is plotted in the opposite direction, which is how inner loops form. Reading $r$ as a function of $\theta$ , exactly as in Cartesian graphing, is the key habit.

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What this topic is asking
Graphing point by point
The standard polar shapes
Negative radius and inner loops
Try this

What this topic is asking

The College Board (Topic 3.14) wants you to graph and interpret a polar function $r = f(\theta)$ , where the radius is a function of the angle. As $\theta$ sweeps around, $r$ changes, tracing a curve. You should recognize the standard polar shapes (circles, rose curves, limacons and spirals) and read features such as where the curve passes through the pole.

Graphing point by point

This is the same input-output table method used for Cartesian graphs, with the twist that the output is a directed distance.

The standard polar shapes

Recognizing the form of the equation tells you the family before you plot a single point, which makes a quick sketch reliable.

Graphing a rose curve

Sketch $r = 3\sin(2\theta)$ and state how many petals it has.

step 1 Identify the family

The form $r = a\sin(n\theta)$ with $n = 2$ is a rose curve. Since $n = 2$ is even, it has $2n = 4$ petals.

step 2 Find where the curve reaches the pole

$r = 0$ when $\sin(2\theta) = 0$ , that is $2\theta = 0, \pi, 2\pi, \ldots$ , so $\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$ . The curve returns to the pole at these angles, separating the petals.

step 3 Find the petal tips

$r$ is largest ( $|r| = 3$ ) when $\sin(2\theta) = \pm 1$ , that is $2\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \ldots$ , so $\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$ . Each gives a petal tip $3$ units from the pole.

step 4 Assemble the graph

Four petals of length $3$ , with tips at $\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$ , meeting at the pole between them. The result is a symmetric four-petal rose.

Negative radius and inner loops

A point worth stating once is what happens when $r$ is negative. A negative radius is plotted in the opposite direction from the angle $\theta$ , along the ray at $\theta + \pi$ . For a limacon such as $r = 1 + 2\cos\theta$ , the radius goes negative for part of the sweep, and those negative- $r$ points trace the inner loop. Tracking the sign of $r$ as $\theta$ increases, rather than assuming $r$ is always a positive distance, is what makes the inner loops and the full shape come out correctly. This sign behavior also drives the rate-of-change analysis of Topic 3.15.

Try this

Q1. What shape is the graph of $r = 5$ ? [1 point]

Cue. A circle of radius $5$ centered at the pole, since the distance is constant for every angle.

Q2. How many petals does $r = 2\cos(3\theta)$ have? [1 point]

Cue. $n = 3$ is odd, so the rose has $n = 3$ petals.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2023 (style)1 marksSection I, Part B (multiple choice, calculator allowed). The graph of the polar function

r = 4\cos\theta

is which shape? (A) A circle (B) A four-petal rose (C) A spiral (D) A line through the origin

Show worked answer →

The correct answer is (A), a circle.

The polar equation $r = 4\cos\theta$ is a standard circle of diameter $4$ passing through the pole, centered on the polar axis. Converting confirms it: multiplying by $r$ gives $r^2 = 4r\cos\theta$ , that is $x^2 + y^2 = 4x$ , or $(x - 2)^2 + y^2 = 4$ , a circle of radius $2$ centered at $(2, 0)$ .

AP 2024 (style)4 marksSection II (free response, calculator allowed). Consider

r = 2 + 2\sin\theta

. (a) Find

r

\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}

. (b) Identify the type of curve and state where it passes through the pole.

Show worked answer →

A 4-point question on reading a polar graph.

(a) Values (2 points): $r(0) = 2 + 2\sin 0 = 2$ ; $r\left(\frac{\pi}{2}\right) = 2 + 2(1) = 4$ ; $r(\pi) = 2 + 2(0) = 2$ ; $r\left(\frac{3\pi}{2}\right) = 2 + 2(-1) = 0$ .
(b) Curve and pole (2 points): since the constant and the coefficient of $\sin\theta$ are equal ( $2 = 2$ ), the curve is a cardioid (a limacon with an inner loop reduced to a cusp). It passes through the pole where $r = 0$ , at $\theta = \frac{3\pi}{2}$ .

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Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)