/*! 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 Graph two periods of each functi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 47

Graph two periods of each function. $$y=\sec \left(2 x+\frac{\pi}{2}\right)-1$$

Short Answer

Expert verified
The graph of the function \(y=\sec \left(2 x+\frac{\pi}{2}\right)-1\) has a period of \(\pi\), it is shifted to the left by \(\frac{\pi}{4}\), and it is shifted downward by \(1\) unit. It should be plotted between \(-\frac{\pi}{4}\) and \( \frac{7\pi}{4}\) for graphing two periods.

Step by step solution

01

Identify the period

The period of a secant function is determined by the coefficient of \(x\) inside the secant function. In our equation, the coefficient of \(x\) is \(2\). The period of the base \(sec(x)\) function is \(2\pi\), therefore, the period of \(sec(2x)\) is \(\pi\). So, one complete cycle of this function will occur over the interval \(0\) to \(\pi\). To get two periods, we must graph from \(0\) to \(2\pi\).
02

Identify the phase shift (or horizontal shift)

The phase shift is determined by the term added or subtracted from \(x\) inside the secant function. In our equation, \( \frac{\pi}{2}\) is added to \(2x\), shifting the graph to the left by \(\frac{\pi}{4}\) (divide by the coefficient of \(x\) to obtain phase shift). Thus, our graphing interval will be \(-\frac{\pi}{4}\) to \( \frac{7\pi}{4}\) (shifting \(0\) to \(2\pi\) to the left by \(\frac{\pi}{4}\))
03

Identify the vertical shift

The vertical shift is determined by the term added or subtracted from the secant function. In our equation, \(1\) is subtracted, shifting the graph down by \(1\) unit. Therefore, the secant vertical asymptote from the base function will shift down by \(1\).
04

Plot the function

Now, refer to the standard secant graph (or graph of \(y=cos(x)\) and then perform inverse function for reference) and plot the function from \(-\frac{\pi}{4}\) to \( \frac{7\pi}{4}\) considering the vertical shift down by \(1\) unit and vertical asymptotes shifted correspondingly.

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

The angular velocity of a point on Earth is \(\frac{\pi}{12}\) radian per hour. Describe what happens every 24 hours.

Use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$y=\tan \frac{x}{4}$$

Use a graphing utility to graph each pair of functions in the same viewing rectangle. Use a viewing rectangle that shows the graphs for at least two periods. $$y=-3.5 \cos \left(\pi x-\frac{\pi}{6}\right) \text { and } y=-3.5 \sec \left(\pi x-\frac{\pi}{6}\right)$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the graph of \(y=3 \cos 2 x\) to obtain the graph of \(y=3 \csc 2 x\)

Use a vertical shift to graph one period of the function. $$y=\cos x+3$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Decision 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.