/*! 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 26 Determine the period of each fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 26

Determine the period of each function. $$ y=\sec (2 \pi x) $$

Short Answer

Expert verified
The period of \(y=\sec(2 \pi x)\) is 1.

Step by step solution

01

Identify the Base Function

The given function is \(y=\sec(2 \pi x)\). Start by identifying the base trigonometric function, which is \(\sec(x)\). The basic period of \(\sec(x)\) is \(2 \pi\).
02

Determine the Coefficient of x

Look at the argument of the secant function in \(y=\sec(2 \pi x)\). The coefficient of \(x\) inside the trigonometric function is \(2 \pi\).
03

Calculate the Period

The period of a function \(y=\sec(bx)\) is calculated using the formula \(\frac{2 \pi}{|b|}\). For this function, \(b=2 \pi\). Thus, the period is \(\frac{2 \pi}{2 \pi} = 1\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

secant function
The secant function, denoted as \(\text{sec}(x)\), is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the cosine function: \(\text{sec}(x) = \frac{1}{\text{cos}(x)}\). This means that wherever \(\text{cos}(x)\) is zero, the secant function becomes undefined, resulting in vertical asymptotes. The secant function forms continuous waves but with gaps where it is undefined. Its range includes all real numbers except those between \(-1\) and \(+1\) inclusive. The basic period of the secant function is \(\text{2蟺}\), meaning the pattern of secant values repeats every \(\text{2蟺}\).
trigonometric period
Trigonometric functions exhibit periodicity, meaning that their values repeat at regular intervals. This property is known as the period. For the standard secant function, \(\text{sec}(x)\), the period is \(\text{2蟺}\). To find the period of modified trigonometric functions like \(\text{sec}(bx)\), one must adjust for the coefficient \(\text{b}\). The period, denoted as \[T\], is calculated using the formula: \[ T = \frac{2蟺}{|b|} \]. In our example \(\text{y=sec(2蟺x)}\), \(\text{b}\) is \(\text{2蟺}\). Plugging this into the formula gives us a period of \(\frac{2蟺}{2蟺} = 1\). This means that the function \(\text{sec}(2蟺x)\) repeats its values every 1 unit along the x-axis.
function transformation
Transformations in trigonometric functions include shifts, stretches, compressions, and reflections. In \(\text{sec}(bx)\), the coefficient \(\text{b}\) applies a horizontal stretch or compression by changing the function鈥檚 period. For \(\text{y=sec(2蟺x)}\), \(\text{b}\) equals \(\text{2蟺}\), which compresses the standard period from \(\text{2蟺}\) to \(\text{1}\). Simply put: a larger \(\text{b}\) reduces the period, squeezing the function's wave to repeat more frequently. Conversely, a smaller \(\text{b}\) expands the period, making the wave repeat less frequently. Understanding these transformations helps in graphing and analyzing functions in various forms.

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

Find the equation of each sine wave in its final position. The graph of \(y=\cos (x)\) is shifted \(\pi / 8\) units to the right, reflected in the \(x\) -axis, and then translated upward 2 units.

For the past three years, the manager of The Toggery Shop has observed that revenue reaches a high of about \(\$ 40,000\) in December and a low of about \(\$ 10,000\) in June, and that a graph of the revenue looks like a sinusoid. If the months are numbered 1 through 36 with 1 corresponding to January, then what are the period, amplitude, and phase shift for this sinusoid? What is the vertical translation? Write a formula for the curve and find the approximate revenue for April.

Let \(f(x)=\tan (x), g(x)=x+3,\) and \(h(x)=2 x .\) Find the following. $$ g(f(h(\pi / 6))) $$

Sketch at least one cycle of the graph of each function. Determine the period and the equations of the vertical asymptotes. $$ y=\cot (3 x+\pi)+2 $$

WRITING/DISCUSSION The Role of the Parameters The numbers \(A, C,\) and \(D\) in \(y=A \sin (x-C)+D\) are called parameters. Explain how each of these parameters affects the location or shape of the graph.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Discrete 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.