Problem 31 Sketch one full period of the gr... [FREE SOLUTION]

Chapter 4: Problem 31

Sketch one full period of the graph of each function. $$y=-3 \cot \frac{x}{2}$$

Short Answer

Expert verified

The graph of the function $y = -3 \cot \frac{x}{2}$ starts and ends at asymptotes at $x=0$ and $x=2\pi$, respectively, and reaches a minimum peak at $x=\pi$, $y=-3$. This pattern will be repeated for other periods.

Step by step solution

Identify the Amplitude and Period

In the function $y = -3 \cot \frac{x}{2}$, the amplitude of the cotangent function is represented by $a = -3$ and the function has a period of $T=\frac{\pi}{|b|} = \frac{\pi}{|\frac{1}{2}|} = 2\pi$.

Identify the Asymptotes

A cotangent function has vertical asymptotes when its argument is a multiple of $\pi$. Because of the coefficient $\frac{1}{2}$ in the argument, the asymptotes for the function are at multiples of $2\pi$. Hence, the vertical asymptotes are at $x = n \cdot 2\pi$, where $n$ is any integer.

Sketching the Graph

Since it's cotangent function, start at one of asymptotes and reach a maximum negative peak (since the function is flipped) halfway to the next asymptote, and return to the next asymptote. From there, copy the same pattern for remaining graph.

Identify the Key Points for Full Period

Some key points for the graph during one period are: $x=0$ where it begins at an asymptote, $x=\pi$ where it hits its lowest point -3, and $x=2\pi$ where it ends at another asymptote. Those are enough to sketch the graph of a single period of the function.

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!

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Sketch one full period of the graph of each function. $$y=-3 \cot \frac{x}{2}$$

Short Answer

Step by step solution

Identify the Amplitude and Period

Identify the Asymptotes

Sketching the Graph

Identify the Key Points for Full Period

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Logic and Functions

Decision Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.