### Understanding Polar Graphs in MathematicsWhen studying polar graphs, it is crucial to understand the shapes and behaviors that different equations can represent. Below, we have two distinct polar graphs which are helpful for understanding various mathematical principles.**Graph B:**- This graph shows a red curve plotted in a polar coordinate system.- The horizontal axis ranges from -4 to 3, while the vertical axis ranges from -3 to 3.- The curve resembles a spiral, starting from the origin and expanding outward. Specifically, the curve seems to loop once around before expanding in a spiral manner. **Graph F:**- This graph also features a red curve plotted in a polar coordinate system.- The horizontal axis ranges from -3 to 3, and the vertical axis ranges from -3 to 3.- The curve forms a single enclosed loop, similar to a circle, centered around the origin. It appears to be a perfect circle with a radius of approximately 2 units.These graphs are useful in visualizing and understanding the complex relationships described by polar equations. In particular, Graph B might represent a type of spiral function, while Graph F is likely showing a circular equation in polar form.**Note:** You can earn partial credit on this problem.By studying these types of graphs, students can better understand the practical applications and visual interpretations of polar coordinates and equations.---_Source: Educational website content_ ### Educational Content: Matching Polar Equations to Plots#### Task OverviewBelow you are given six polar equations and plots of four of these curves. Match each plot with the correct equation. (Note: Values along the axes are given in Cartesian coordinates.)#### Given Polar EquationsA. \(\theta = -\pi/6\)B. \(r^2 - 3r + 2 = 0\)C. \(r = \sin(\theta)\)D. \(r = -3\cos(\theta)\)E. \(r = 2\left(1 - \sin(\theta)\right)\)F. \(r = 1 - 3\cos(\theta)\)#### Graphical Plots Explained1. **First Graph (Label D):** - Description: The graph shows a heart-shaped curve oriented along the horizontal axis, with its center at the origin. It spans from -4 to 4 on the x-axis and from -4 to 1 on the y-axis. - Equation: \(r = 2\left(1 - \sin(\theta)\right)\)2. **Second Graph (Label A):** - Description: The graph shows a circle centered at the origin. The circle has a radius of 3 units. - Equation: \(r = -3\cos(\theta)\)#### Analysis of Graphical Plots:- In the first graph (Label D), the shape of the plot is indicative of a polar equation that modifies both the radial and angular components in a sinusoidal manner. The heart-like shape (often referred to as a cardioid) aligns with the given equation \(r = 2\left(1 - \sin(\theta)\right)\).- In the second graph (Label A), the circular shape (a conic section circle) is centered at the origin and reveals a standard form of a circle equation in polar form, specifically with \(r\) being proportional to \(\cos(\theta)\). The equation \(r = -3\cos(\theta)\) correctly represents this plot, indicating a radius of 3 units but mirrored horizontally.This transcription includes the task description and detailed explanations of the graphs provided.

Answered: Image /qna-images/answer/fb02ab63-d633-4cfd-9cbf-f5efdf00d3d5.jpg

Answered: B F 3. 4. NOO ข่ง 3

B F 3. 4. NOO ข่ง 3

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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