/*! 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 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 46

Sketch the graph of the function. (Include two full periods.) $$y=\sin \frac{\pi x}{4}$$

Short Answer

Expert verified
The graph of the function \(y=\sin \frac{\pi x}{4}\) is a sine wave with a period of 8 units. It starts at 0 at the origin, peaks at x = 2, crosses the x-axis again at x = 4, reaches a negative peak at x = 6, and returns to 0 at x = 8. This pattern repeats every 8 units on the x-axis.

Step by step solution

01

Define the Function

The function is defined as \(y=\sin \frac{\pi x}{4}\). The standard form for a sine function is \(y = A \sin (Bx - C) + D\), where B affects the period.
02

Find the Period of Function

The standard period of a sine function is \(2\pi\). Since the coefficient of \(x\) in our function is \(\pi/4\), we need to divide the standard period by this coefficient to get the period for our function. So the period for our function is \(2\pi/(\pi/4) = 8\). The function will repeat itself every 8 units on the x-axis.
03

Plot Key Points

For a sine function, key points within a period occur at \(0, T/4, T/2, 3T/4, T\), where \(T\) is the period of function. So for this function, key points occur at \(0, 8/4, 8/2, 24/4, 8\) = \(0,2,4,6,8\). Use these x-values and the y-values obtained by evaluating the function at these points to plot the key points.
04

Sketch the Graph

After determining the key points, plot them on a set of axes. Use more periods as needed to create the sketch. The curve should begin at 0, reach a peak at \(T/4\) (2), go back down to 0 at \(T/2\) (4), reach a negative peak at \(3T/4\) (6), and go back to 0 at \(T\) (8). The process then repeats for subsequent 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

Determine whether the statement is true or false. Justify your answer. The Leaning Tower of Pisa is not vertical, but when you know the angle of elevation \(\theta\) to the top of the tower as you stand \(d\) feet away from it, you can find its height \(h\) using the formula \(h=d \tan \theta\)

A ship leaves port at noon and has a bearing of \(\mathrm{S} 29^{\circ} \mathrm{W}\). The ship sails at 20 knots. (a) How many nautical miles south and how many nautical miles west will the ship have traveled by 6: 00 P.M.? (b) At 6: 00 e.m., the ship changes course to due west. Find the ship's bearing and distance from the port of departure at 7: 00 P.M.

Determine whether the statement is true or false. Justify your answer. $$\tan \frac{5 \pi}{4}=1 \quad \rightarrow \quad \arctan 1=\frac{5 \pi}{4}$$

Sketch a graph of the function. $$y=2 \arccos x$$

Use a graphing utility to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as \(x\) increases without bound. $$h(x)=2^{-x^{2} / 4} \sin x$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

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.