/*! 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 30 Sketch one full period of the gr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 30

Sketch one full period of the graph of each function. $$y=-3 \tan 3 x$$

Short Answer

Expert verified
The graph of \(y=-3 \tan 3x\) is a reflection of the standard tangent function about the x-axis, stretched vertically by a factor of 3, with vertical asymptotes at \(x=0\) and \(x=\pi/3\). The function approaches these asymptotes, passing through the point (\(\pi/6\), 0).

Step by step solution

01

Identify Amplitude, Period, Phase Shift and Vertical Shift

The general form of a tangent function is \(y=A \tan(Bx-C)+D\), where A is the amplitude, B affects the period, C the phase shift, and D the vertical shift. For the given function, \(y=-3 \tan 3x\), A=-3, B=3, and there is no phase shift (C=0) or vertical shift (D=0). Therefore, the amplitude is 3 (since amplitude is the absolute value of A), the period is \(\pi/B=\pi/3\), and this function is a reflection of the standard tangent function with a vertical stretch by a factor of 3.
02

Plot key points

Since the period of the function is \(\pi/3\), first plot vertical asymptotes at x=0 and at x=\(\pi/3\). Next, find the mid-point of the asymptotes, which, in this case, is \(\pi/6\). For the tangent function, the value at this mid-point is 0. However, due to the reflection and vertical stretch, the function value is -3*0=0.
03

Sketch the Graph

Between the asymptotes, the function will start at the first asymptote, pass through the point (\(\pi/6\), 0), and then approach the second asymptote. The tangent function is negative as it reflects across the x-axis, implying it starts from the first asymptote, goes downwards to the mid-point, then goes upwards approaching the next asymptote. Continue plotting more periods if necessary, using the period to find the location of asymptotes and mid-points.

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

Graph at least one full period of the function defined by each equation. $$y=-\left|2 \sin \frac{x}{3}\right|$$

The average high temperature \(T,\) in degrees Fahrenheit, for Fairbanks, Alaska, is given by $$ T(t)=-41 \cos \left(\frac{\pi}{6} t\right)+36 $$ where \(t\) is the number of months after January \(5 .\) Use the formula to estimate (to the nearest 0.1 degree Fahrenheit) the average high temperature in Fairbanks for March 5 and July 20.

Use a graphing utility to graph each function. $$y=\cos ^{2} x$$

Use the fundamental trigonometric identities to find the value of the function. Given \(\csc t=\sqrt{2}, 0

Explain how to use the graph of \(y=f(x)\) to produce the graph of \(y=-f(x)\) .
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.