/*! 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 47 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 47

Use a graphing utility to graph the function. (Include two full periods.) $$y=0.1 \tan \left(\frac{\pi x}{4}+\frac{\pi}{4}\right)$$

Short Answer

Expert verified
The graph of \(y=0.1 \tan \left(\frac{\pi x}{4}+\frac{\pi}{4}\right)\) starts at a center point, moves up to a maximum, returns to the center, and then drops to a minimum. This pattern is repeated for two complete periods. The graph is narrower with a period of \(4\pi\) and is shifted horizontally to the left by 1 unit. It is also vertically stretched by a factor of 0.1, causing it to appear 'shorter' than the typical tangent graph.

Step by step solution

01

Recognize Tangent Function and Its Transformations

The given function is a form of the tangent function \(y = A \tan(Bx - C) + D\) where \(A\) is the amplitude, \(B\) is how the graph is horizontally stretched or compressed, \(C\) is the phase shift, and \(D\) is vertical shift. From the given function \(y=0.1 \tan \left(\frac{\pi x}{4}+\frac{\pi}{4}\right)\), it can be seen that \(A = 0.1\), \(B = \frac{\pi}{4}\), and \(C = \frac{\pi}{4}\). There is no \(D\) hence no vertical shift.
02

Determine Period of Function

The period of a tangent function is determined by the equation \(\frac{2\pi}{|B|}\). Substituting the given \(B\) value, the period of the function is \(4\pi\).
03

Determine the Phase Shift

The Phase shift is determined by the equation \(\frac{C}{|B|}\). Hence, plugging in the given \(C\) and \(B\) values, the phase shift is -1.
04

Graph the Function

Knowing the amplitude, period, and phase shift, plot the function considering that tangent function begins at a center point, goes up to a maximum, returns to the center, and then drops to a minimum. Repeat this pattern for two complete periods. Remember the graph is transformed to be narrower with period \(4\pi\) and horizontally shifted left by 1 unit. It is also stretched vertically by a factor of 0.1

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. $$f(x)=\frac{\pi}{2}+\cos ^{-1}\left(\frac{1}{\pi}\right)$$

Fill in the blank. If not possible, state the reason. $$\text { As } x \rightarrow \infty, \text { the value of } \arctan x \rightarrow\text { _____ } .$$

Consider the functions \(f(x)=\sin x\) and \(f^{-1}(x)=\arcsin x\). (a) Use a graphing utility to graph the composite functions \(f \circ f^{-1}\) and \(f^{-1} \circ f\). (b) Explain why the graphs in part (a) are not the graph of the line \(y=x .\) Why do the graphs of \(f \circ f^{-1}\) and \(f^{-1}\) o \(f\) differ?

A satellite in a circular orbit 1250 kilometers above Earth makes one complete revolution every 110 minutes. Assuming that Earth is a sphere of radius 6378 kilometers, what is the linear speed (in kilometers per minute) of the satellite?

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. $$f(x)=2^{-x / 4} \cos \pi x$$
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Applied Mathematics

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.