/*! 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 59 Use a vertical shift to graph on... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 59

Use a vertical shift to graph one period of the function. $$y=-3 \cos 2 \pi x+2$$

Short Answer

Expert verified
The graph is a vertically shifted cosine curve which starts from bottom because of the negative amplitude at \(y=-1\), peaks at \(y=5\), and has one complete wave in the interval of \(1/2 \pi\) along the x axis.

Step by step solution

01

Identify the amplitude, frequency, and vertical shift

In the function \(y=-3 \cos 2 \pi x+2\), the amplitude is \(-3\), which means the wave will have a height of 3. The frequency is \(2 \pi\), which means one period of the wave will be completed in an interval of \(1/2 \pi\). The vertical shift is \(+2\), which means the whole wave will be moved up by 2 units from the x-axis.
02

Determine the range

The range of the function is determined by the amplitude and the vertical shift. Because the amplitude is 3, the wave extends 3 units above and below its central position. The graph has been moved up by 2 units, so the graph will range from \(2-3=-1\) up to \(2+3=5\).
03

Plot the function

First, draw a dashed line \((y=2)\) to represent the vertical shift of the wave. Then, remember that the cosine function starts at the peak of a wave. Because of the negative amplitude, these waves will instead start at the bottom. Draw a period of the cosine curve, making sure it has a peak at \(y=5\) and a trough at \(y=-1\). Also note that the period of one wave should span \(1/2 \pi\) along the x-axis.

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

Use a vertical shift to graph one period of the function. $$y=2 \cos \frac{1}{2} x+1$$

Graph \(f, g,\) and \(h\) in the same rectangular coordinate system for \(0 \leq x \leq 2 \pi .\) Obtain the graph of h by adding or subtracting the corresponding \(y\) -coordinates on the graphs of \(f\) and \(g\) $$f(x)=2 \cos x, g(x)=\cos 2 x, h(x)=(f+g)(x)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. If \(\theta=\frac{3}{2},\) is this angle larger or smaller than a right angle?

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-3.5 \cos \left(\pi x-\frac{\pi}{6}\right) \text { and } y=-3.5 \sec \left(\pi x-\frac{\pi}{6}\right)$$

Have you ever noticed that we use the vocabulary of angles in everyday speech? Here is an example: My opinion about art museums took a \(180^{\circ}\) turn after visiting the San Francisco Museum of Modern Art. Explain what this means. Then give another example of the vocabulary of angles in everyday use.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.