/*! 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 70 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 70

Use a graphing utility to graph the function. (Include two full periods.) Be sure to choose an appropriate viewing window. $$y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2$$

Short Answer

Expert verified
First, calculate the period of the function, which gives the required viewing window in x-direction. Then identify the amplitude and vertical shift to set the viewing window in the y-direction. Input the function into a graphing utility and set the viewing window as determined. Two full periods of the graph from 0 to 8 on the x-axis and from -5 to 1 on the y-axis should be displayed.

Step by step solution

01

Calculate the period of the cosine function

The period of a cosine function is given by the formula \( \frac{2 \pi}{|b|} \). Here, \( b = \frac{\pi}{2} \). So, the period \( P= \frac{2 \pi}{|\frac{\pi}{2}|} = 4 \). So, we need to include two full periods from 0 to 4*2=8 in the x-axis in the graph. This gives the required viewing window in the x-direction.
02

Identify the Amplitude and Vertical Shift

The amplitude of the cosine function is given by the absolute value of the coefficient of the cosine term, here \( a=3 \). So, the amplitude is 3, which means the graph will reach 3 units above and below its center line. The function also shifts 2 units downwards which is stated by the '-2' at the end of the equation. This means the center line of the function is \( y = -2 \). Considering these facts, the viewing window in the y-direction must include the range from \( -2-3 \) to \( -2+3 \), which is -5 to 1.
03

Graph the function

Using a graphing utility, input the function \(y = 3\cos\left(\frac{\pi x}{2}+\frac{\pi}{2}\right) -2 \) and set the viewing window as identified in the previous steps, i.e. 0 to 8 for x and -5 to 1 for y. Depending on the graphing utility, there might be a different approach to input the function and set the viewing window. Make sure to follow instructions for that specific tool and obtain the graph.

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 graphing utility to graph the function. Use the graph to determine the behavior of the function as \(x \rightarrow c\) (a) \(x \rightarrow\left(\frac{\pi}{2}\right)^{+}\) (b) \(x \rightarrow\left(\frac{\pi}{2}\right)^{-}\) (c) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{+}\) (d) \(x \rightarrow\left(-\frac{\pi}{2}\right)^{-}\) $$f(x)=\tan x$$

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow-1^{+}, \text {the value of } \arccos x \rightarrow\text { _____ } .$$

Use a graphing utility to graph the function. $$f(x)=\pi \arcsin (4 x)$$

Finding the Central Angle Find the radian measure of the central angle of a circle of radius \(r\) that intercepts an are of length \(s\). \(r=14\) feet \(, s=8\) feet

The numbers of hours \(H\) of daylight in Denver, Colorado, on the 15 th of each month are: \(1(9.67), 2(10.72), \quad 3(11.92), \quad 4(13.25)\) \(5(14.37), \quad 6(14.97), \quad 7(14.72), \quad 8(13.77), \quad 9(12.48)\) \(10(11.18), \quad 11(10.00), \quad 12(9.38) . \quad\) The month is represented by \(t,\) with \(t=1\) corresponding to January. A model for the data is \(H(t)=12.13+2.77 \sin \left(\frac{\pi t}{6}-1.60\right)\) (a) Use a graphing utility to graph the data points and the model in the same viewing window. (b) What is the period of the model? Is it what you expected? Explain. (c) What is the amplitude of the model? What does it represent in the context of the problem? Explain.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Probability and Statistics

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.