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Chapter 6: Problem 80

Use a graphing utility to graph and identify \(r=2+k \sin \theta\) for \(k=0,1,2,\) and \(3 .\)

Short Answer

Expert verified
The plots represent a series of cardioid shapes with one 'lobe'. The lobe becomes more pronounced as k increases, while the radius along the x-axis remains constant at 2

Step by step solution

01

Graph for k=0

Set \(k = 0\) in the given function \(r=2+k \sin \theta\). The function becomes \(r = 2+0 \sin \theta\) which simplifies to \(r = 2\). This is a circle with radius 2, centered at the origin.
02

Graph for k=1

Set \(k = 1\) in the function. The function becomes \(r = 2+ 1\sin \theta\), or \(r = 2+\sin \theta\). Plot this on your graphing utility and observe the change from the original graph.
03

Graph for k=2

With \(k = 2\), the function changes to \(r = 2+ 2\sin \theta\), or \(r = 2+2\sin \theta\). Plot this on your graphing utility and observe the change from the previous graphs.
04

Graph for k=3

Replacing \(k\) with 3 gives \(r = 2+ 3\sin \theta\). Plot this on your graphing utility and observe the change from the previous graphs.
05

Identification

By comparing the graphs, identify and note the effects of changing k values on the polar function. As the value of k increases, the 'lobe' on the positive y-axis becomes more pronounced. The radius of the section along the x-axis remains at 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphing Utility
A graphing utility is a powerful tool that helps us visualize mathematical equations and their transformations. Since polar equations can be a bit tricky to plot manually, using a graphing utility becomes essential. Polar graphs use a coordinate system where each point is determined by a distance from the pole (the origin) and an angle from the polar axis.

When using a graphing utility:
  • You simply input the equation; it handles the plotting.
  • It offers capabilities to adjust the angle increments for smoother graphs.
  • You can zoom in or out to better see the effects and shape.
For the specific exercise with the equation \( r = 2 + k \sin \theta \), entering different values of \( k \) will allow you to observe how the graph changes shape. This visual aid makes understanding the behavior of polar graphs much easier.
Effects of Parameters
The effects of parameters in an equation like \( r = 2 + k \sin \theta \) are crucial in shaping the graph. The parameter \( k \) in this function modulates the influence of the trigonometric part on the radius of the polar circle:

  • When \( k = 0 \), there is no modulation from the sine function; resulting in a perfect circle with radius 2.
  • Increasing \( k \) introduces variations caused by the \( \sin \theta \) term.
  • A higher \( k \) value enhances the amplitude of these variations, creating more distinct shapes, like lobes.
Visualizing these effects using a graphing utility makes it clear how changing \( k \) alters the graph's appearance. As \( k \) increases, features become more pronounced, illustrating vivid alterations on the y-axis.
Circle Equation
In polar coordinates, a circle can be represented in a simple form. The constant term in the equation \( r = 2 + k \sin \theta \) contributes to the circle's base size. For \( k = 0 \), the equation reduces to \( r = 2 \), representing a circle with a radius of 2 centered at the pole. This is akin to the Cartesian equation of a circle, such as \( x^2 + y^2 = r^2 \).

However, adding a trigonometric component shifts or distorts the circle. Here, it leads to bulging on parts of the circle, creating a new shape depending on \( k \). Understanding this helps in identifying how circles and their derivatives appear in polar graphs.
Trigonometric Functions
Trigonometric functions, like sine and cosine, play a significant role in polar graphs. In our equation \( r = 2 + k \sin \theta \), \( \sin \theta \) affects the radial distance for different values of \( \theta \). It oscillates between -1 and 1, modulating the radius around the constant value.

  • The sine component affects the position of the graph with respect to the polar axis.
  • The behavior of \( \sin \theta \) at different \( \theta \) values helps create familiar polar shapes like cardioids or more complex inner loops.
  • As \( k \) changes, it amplifies or reduces the impact of \( \sin \theta \), visually transforming the graph.
Understanding how trigonometric functions interact with polar equations enhances comprehension of how such graphs are constructed and interpreted.

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Most popular questions from this chapter

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Use a graphing utility to graph the ellipse. Find the center, foci, and vertices. (Recall that it may be necessary to solve the equation for \(y\) and obtain two equations.) \(5 x^{2}+3 y^{2}=15\)
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