/*! 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 46 Graph two periods of each functi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 46

Graph two periods of each function. $$y=2 \cot \left(x+\frac{\pi}{6}\right)-1$$

Short Answer

Expert verified
The graph of `y = 2 cot(x + \pi/6) - 1` will show a repeated cotangent curve, starting from the phase shift of \(-\pi/6\), going upwards through the mid-point between asymptotes at \( y = -1 \), and continuing to the next asymptote. This pattern repeats every \( \pi \) units along the x-axis, and will be shifted downwards by 1 unit.

Step by step solution

01

Identifying Key Elements

We start by identifying key elements from the given function. The function is y = 2 cot(x + \pi/6) - 1. The amplitude is not applicable here. The period can be found by dividing \( \pi \) by the absolute value of 'b'. From the function, 'b' is 1, so the period is \( \pi \ / \ |1| \ = \ \pi \). The phase shift can be calculated as '-c/b'. Here, 'c' is \(\pi/6\) and 'b' is 1, so the phase shift is \(-\pi/6\). The vertical shift 'd' is -1, which means the function is shifted downward by 1 unit.
02

Plotting the Asymptotes

We know that cotangent function has vertical asymptotes wherever the function is undefined i.e., at every multiplier of \( \pi \ + \ phase \ shift \). So the asymptotes will be located at multiples of \( \pi \ + \ -\pi/6 \). Therefore asymptotes will be located at \( -\pi/6 \ , \ 5\pi/6 \ , \ 3\pi/2 \)... Plot these asymptotes on the graph first.
03

Plotting the Key Points

Plot the mid-point between each set of asymptotes. These represent the 'zeroes' of the cotangent function, where the function intersects the x-axis. It will be at the middle points between the asymptotes i.e., at \( \pi/6 \ , \ 4\pi/6 \ , \ 7\pi/6 \)..., and the corresponding 'y' values will be equal to the value of vertical shift, which is '-1'. Plot these on the graph. Next, plot more points by taking the middle of each half period. The corresponding 'y' values will vary depending on whether it is a maximum or minimum. In our case, all maxima points will take the value 1, and minima will take -3.
04

Graphing the Function

Start at the point representing the phase shift, and make sure your function approaches asymptotes as x approaches the x-values of the asymptotes. The function will stay in-between two vertical asymptotes for every period. Make sure the function is periodic over the distance of the period calculated in step 1, and that it covers two full periods.

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

Most popular questions from this chapter

The following figure shows the depth of water at the end of a boat dock. The depth is 6 feet at low tide and 12 feet at high tide. On a certain day, low tide occurs at 6 A.M. and high tide at noon. If \(y\) represents the depth of the water \(x\) hours after midnight, use a cosine function of the form \(y=A \cos B x+D\) to model the water's depth.

Describe a general procedure for obtaining the graph of \(y=A \sin (B x-C)\)

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Determine the range of each of the following functions. Then give a viewing rectangle, or window, that shows two periods of the function's graph. a. \(f(x)=3 \sin \left(x+\frac{\pi}{6}\right)-2\) b. \(g(x)=\sin 3\left(x+\frac{\pi}{6}\right)-2\)

Solve: \(\log _{3}(x+5)=2\) (Section 3.4, Example 6)

Graph one period of each function. $$y=-\left|2 \sin \frac{\pi x}{2}\right|$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

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.