/*! 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 Graph one period of each functio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 70

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

Short Answer

Expert verified
The graph of the function \(y=-\left|2 \sin \frac{\pi x}{2}\right|\) is an inverted sine curve starting from \(x=0\) to \(x=4\) on the x-axis, oscillating downwards due to the absolute value and the negative value.

Step by step solution

01

Understand the function

The student is dealing with a sine function which is multiplied by 2, has an absolute value, and its outcome is furthermore negated. The \(\frac{\pi x}{2}\) part inside the sine function also affects the period of the sine wave. The negative sign at the front of the absolute value function flips it downwards.
02

Identify the period of the function

The period depends upon the coefficient of \(x\) inside the sine function. Normally, the period of \(\sin(x)\) is \(2\pi\). If we have a coefficient \(b\) multiplied with \(x\), the period becomes \(\frac{2\pi }{ b}\). Here, the coefficient of \(x\) is \(\frac{\pi}{2}\), so the period of the function is \(\frac{2\pi }{ \frac{\pi}{2}} = 4\). So, the graph of the function repeats after every 4 units.
03

Calculate key points for one period and plot

We can calculate key points in all significant places in a period (start - mid - end points as well as tips). Starting from \(x=0\) to \(x=4\), calculate \(y\) values for \(x = 0, 1, 2, 3, 4\) and plot these points. Note that the absolute value of \(y\) will always be positive, but due to the negative sign, the graph will lie below the x-axis. Thus, our graph will be an inverted wave under the x-axis from \(x = 0\) to \(x = 4\).
04

Draw the graph of the function

Finally, draw the graph using these key points. Make sure you just draw one period of the function, from \(x=0\) to \(x=4\). The graph should oscillate below the x-axis, reflecting the fact that the function is the absolute value of a trigonometric function and it's negative.

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

What determines the size of an angle?

Graph \(y=\tan ^{-1} x\) and its two horizontal asymptotes in a \([-3,3,1]\) by \(\left[-\pi, \pi, \frac{\pi}{2}\right]\) viewing rectangle. Then change the viewing rectangle to \([-50,50,5]\) by \(\left[-\pi, \pi, \frac{\pi}{2}\right] .\) What do you observe?

Explain why, without restrictions, no trigonometric function has an inverse function.

Without drawing a graph, describe the behavior of the graph of \(y=\sin ^{-1} x .\) Mention the function's domain and range in your description.

Use a graphing utility to graph two periodsof the function. Use a graphing utility to graph \(y=\sin x+\frac{\sin 2 x}{2}+\frac{\sin 3 x}{3}+\frac{\sin 4 x}{4}\) in a \(\left[-2 \pi, 2 \pi, \frac{\pi}{2}\right]\) by \([-2,2,1]\) viewing rectangle. How do these waves compare to the smooth rolling waves of the basic sine curve?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

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.