Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Question
Sketch a graph for the equation
![**Equation:**
\[ r = 3 \sin(2\theta) \]
**Graph Description:**
This is a polar graph representing the equation \( r = 3 \sin(2\theta) \). It's depicted on a polar coordinate system, which is composed of concentric circles centered at the origin and radiating lines that intersect these circles at various angles.
**Details:**
- **Concentric Circles:** The graph shows circles marked at intervals, typically representing the radius values \(1, 2, 3,\) and so on. Each circle corresponds to a constant radius value.
- **Radial Lines:** The lines radiate from the center at different angles, marked in radians (\(0, \pi/4, \pi/2, 3\pi/4, \pi, 5\pi/4, 3\pi/2, 7\pi/4\)). These help indicate the angle \(\theta\).
- **Plot of \( r = 3 \sin(2\theta) \):** The graph displays a four-petaled rose curve. This type of curve results from the sinusoidal function with a doubled angle \((2\theta)\). The number of petals is twice the coefficient of \(\theta\) within the sine function.
- **Symmetry:** The curve is symmetric about the origin, and each petal spans an angle of \(\pi/2\) radians. The maximum radius (3) occurs at certain angles determined by the function.
This graph and equation are part of the study of polar coordinates and polar equations, illustrating the elegance and variety of shapes that can be graphed with sinusoidal functions in the polar system.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6a85ba4d-6f5c-41bf-97e5-182adb23b838%2Fbef64cee-b3cd-41b7-91fd-729197495724%2Ftwl99jt_processed.png&w=3840&q=75)
Transcribed Image Text:**Equation:**
\[ r = 3 \sin(2\theta) \]
**Graph Description:**
This is a polar graph representing the equation \( r = 3 \sin(2\theta) \). It's depicted on a polar coordinate system, which is composed of concentric circles centered at the origin and radiating lines that intersect these circles at various angles.
**Details:**
- **Concentric Circles:** The graph shows circles marked at intervals, typically representing the radius values \(1, 2, 3,\) and so on. Each circle corresponds to a constant radius value.
- **Radial Lines:** The lines radiate from the center at different angles, marked in radians (\(0, \pi/4, \pi/2, 3\pi/4, \pi, 5\pi/4, 3\pi/2, 7\pi/4\)). These help indicate the angle \(\theta\).
- **Plot of \( r = 3 \sin(2\theta) \):** The graph displays a four-petaled rose curve. This type of curve results from the sinusoidal function with a doubled angle \((2\theta)\). The number of petals is twice the coefficient of \(\theta\) within the sine function.
- **Symmetry:** The curve is symmetric about the origin, and each petal spans an angle of \(\pi/2\) radians. The maximum radius (3) occurs at certain angles determined by the function.
This graph and equation are part of the study of polar coordinates and polar equations, illustrating the elegance and variety of shapes that can be graphed with sinusoidal functions in the polar system.
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