/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 32 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 32

Use a graphing utility to graph each equation. $$r=-\theta, 0 \leq \theta \leq 6 \pi$$

Short Answer

Expert verified
The graph of the polar equation \(r=-\theta\) is a spiral starting from the origin and expanding out in a counter-clockwise direction into the negative quadrants.

Step by step solution

01

Understanding the polar equation

The given equation is a polar equation. In polar coordinates, a point in the plane is described by its distance from the origin (denoted as \(r\)) and the angle it makes with the positive x-axis (denoted as \(\theta\)). In the equation \(r=-\theta\), as \(\theta\) increases, \(r\) becomes more and more negative which indicates that it's moving in the opposite direction.
02

Inputting the equation in the graphing utility

Input the polar equation \(r=-\theta\) into a graphing utility. Make sure that the graphing mode is set to 'Polar'. The domain of \(\theta\) is between 0 and \(6 \pi\). Often, the domain of the graphing utility is already set to an appropriate range, this is checked to ensure it includes all necessary values.
03

Graphing the equation

Press the 'GRAPH' button to plot the equation. The graph can be seen as a spiral starting from the origin and expanding outwards in a counter-clockwise direction. With every increasing loop, the spiral proceeds further into the negative quadrants.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Graphing Utilities
A graphing utility is a tool, often software or a calculator, that helps visualize equations and data. With one, you can easily plot graphs and interpret mathematical equations. Here's how they work:
  • Types: They might be built into calculators or available as computer software. Popular software includes Desmos and GeoGebra.
  • Uses: They handle different types of graphs, such as Cartesian or polar. You can input equations and immediately see a visual output.
  • Benefits: These tools save time and improve accuracy. They also aid in learning by providing visual aids.
When graphing polar equations, set the mode to 'Polar' to ensure correct graphing. The tool will interpret angles and radial distances appropriately. This setting is crucial for observing the unique characteristics of polar graphs.
Exploring Polar Equations
Polar equations differ from Cartesian equations by using radius and angle. In these equations, points are defined as \((r, \theta)\), where:
  • \(r\) is the distance from the origin.
  • \(\theta\) is the angle with the positive x-axis.
The equation \(r=-\theta\) means that the radius \(r\) is the negative of the angle \(\theta\). As \(\theta\) increases within the range \([0, 6\pi]\), the radius becomes more negative, causing the point to spiral outwards in the opposite direction. This behavior results in a distinct and interesting shape known as a spiral.
Graphing Polar Equations
When graphing polar equations, like \(r=-\theta\), understanding the output involves looking for patterns and shapes. Here's how to approach graphing:
  • Mode: Ensure your graphing utility is set to 'Polar'. This mode calculates the position based on angles and radii.
  • Input: Enter the polar equation carefully. For \(r=-\theta\), ensure values fit within the specified range \([0, 6\pi]\).
  • Observation: The resulting graph resembles a spiral. It starts at the origin and unwinds counter-clockwise, moving through negative quadrants as \(r\) becomes more negative.
This method allows for visualizing complex mathematical relationships and deepens understanding of how polar coordinates map onto a plane.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the sum of the infinite geometric series. $$\sum_{n=1}^{\infty}\left(-\frac{3}{5}\right)^{n}$$

Use the Multinomial Theorem to find the indicated coefficient. Find the coefficient of \(a^{5} b^{2} c^{2}\) in the expansion of \((a+b+c)^{9}\)

Find the \(n\) th term of the geometric sequence. $$0.5,0.05,0.005, \ldots$$

The graph of $$r=1.5^{\sin \theta}-2.5 \cos 4 \theta+\sin ^{7} \frac{\theta}{15}$$ is a butterfly curve similar to the one shown below. (GRAPH CAN'T COPY) Use a graphing utility to graph the butterfly curve for a. \(0 \leq \theta \leq 5 \pi\) b. \(0 \leq \theta \leq 20 \pi\) For additional information on butterfly curves, read "The Butterfly Curve" by Temple H. Fay, The American Mathematical Monthly, vol. \(96,\) no. 5 (May 1989 ), \(\mathrm{p} .442\)

Write each rational number as the quotient of two integers in simplest form. $$0 . \overline{355}$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.