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Sketch an excellent graph for the following equatic a) r = 2 cos(0) + 1 3π/4 TT/2 21 31 π/4

Sketch an excellent graph for the following equatic a) r = 2 cos(0) + 1 3π/4 TT/2 21 31 π/4

Calculus: Early Transcendentals
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...
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### Polar Graph Analysis **Objective:** To sketch and analyze the graph for the polar equation \( r = 2 \cos(\theta) + 1 \). **Graph Description:** The graph depicted is a polar plot. The polar coordinate system is used, where each point on the plane is determined by a distance \( r \) from the origin and an angle \( \theta \) from the horizontal axis. **Equation Analysis:** - The given equation is \( r = 2 \cos(\theta) + 1 \). - This represents a polar equation where \( r \) changes based on the cosine of the angle \( \theta \). **Plot Explanation:** 1. **Axes and Grid:** - The graph is circular, centered at the origin, with concentric circles around it. - The circles represent different radii at regular intervals. - The grid has radial lines at angles \( 0, \pi/4, \pi/2, 3\pi/4, \pi, 5\pi/4, 3\pi/2, \text{and } 7\pi/4 \). 2. **Graphing the Equation:** - Start with \( \theta = 0 \), \( r = 2 \cos(0) + 1 = 3 \). - At \( \theta = \pi/2 \), \( r = 2 \cos(\pi/2) + 1 = 1 \). - At \( \theta = \pi \), \( r = 2 \cos(\pi) + 1 = -1 \). - These points show how \( r \) changes with \( \theta \), producing a cardioid-like shape. 3. **Key Features:** - The maximum radius is 3, corresponding to \( \theta = 0 \). - The shape created by this equation is a cardioid, a common polar graph form that typically results when equations involve cosine and sine. - The cardioid has a boundary touching the origin point on the left half of the plot at a minimum point when \( r \) becomes negative and reverts to positive at opposite angles. This polar plot is a visualization tool that helps understand how the variable \( r \) behaves based on the angle \( \theta \), forming complex and symmetric shapes like the cardioid seen here.
Transcribed Image Text:### Polar Graph Analysis **Objective:** To sketch and analyze the graph for the polar equation \( r = 2 \cos(\theta) + 1 \). **Graph Description:** The graph depicted is a polar plot. The polar coordinate system is used, where each point on the plane is determined by a distance \( r \) from the origin and an angle \( \theta \) from the horizontal axis. **Equation Analysis:** - The given equation is \( r = 2 \cos(\theta) + 1 \). - This represents a polar equation where \( r \) changes based on the cosine of the angle \( \theta \). **Plot Explanation:** 1. **Axes and Grid:** - The graph is circular, centered at the origin, with concentric circles around it. - The circles represent different radii at regular intervals. - The grid has radial lines at angles \( 0, \pi/4, \pi/2, 3\pi/4, \pi, 5\pi/4, 3\pi/2, \text{and } 7\pi/4 \). 2. **Graphing the Equation:** - Start with \( \theta = 0 \), \( r = 2 \cos(0) + 1 = 3 \). - At \( \theta = \pi/2 \), \( r = 2 \cos(\pi/2) + 1 = 1 \). - At \( \theta = \pi \), \( r = 2 \cos(\pi) + 1 = -1 \). - These points show how \( r \) changes with \( \theta \), producing a cardioid-like shape. 3. **Key Features:** - The maximum radius is 3, corresponding to \( \theta = 0 \). - The shape created by this equation is a cardioid, a common polar graph form that typically results when equations involve cosine and sine. - The cardioid has a boundary touching the origin point on the left half of the plot at a minimum point when \( r \) becomes negative and reverts to positive at opposite angles. This polar plot is a visualization tool that helps understand how the variable \( r \) behaves based on the angle \( \theta \), forming complex and symmetric shapes like the cardioid seen here.
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