/*! 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 40 Graph two periods of the given c... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 40

Graph two periods of the given cosecant or secant function. $$y=-\frac{3}{2} \sec \pi x$$

Short Answer

Expert verified
The graph of the function \(y=-\frac{3}{2} \sec \pi x\) would consist of upside-down U-shaped curves due to the negative reflection and the 3/2 vertical stretch. The vertical asymptotes are at \(x=\frac{1}{2}+k\) for any integer \(k\). The curve repeats after every period of 1.

Step by step solution

01

Draw the basic function graph

The basic shape of the \(\sec x\) function is a series of U-shaped curves with vertical asymptotes at \(x=\pi/2 + k\pi\) for any integer \(k\). The maximum and minimum points are at \(y=1\) and \(y=-1\) respectively for \(\sec x\) function.
02

Reflect and Stretch the Graph

The negative sign in front of the function indicates a reflection along the x-axis. The absolute value of the coefficient of secant (3/2 here) stretch the graph vertically by a factor of 3/2.
03

Apply the horizontal shift

The function is \(\sec \pi x\), where \(\pi\) is the coefficient of \(x\). This results in multiplying the x-values by \(1/\pi\), effectively compressing the graph horizontally. We incorporate this into the range of the function.
04

Graph

Use the asymptotes as guides, draw the reflected and stretched \(\sec \pi x\) function through one full cycle. As this exercise required the creation of two periods of the function, repeat the cycle immediately after the first to complete the graph.

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 \( \sec x \), is the reciprocal of the cosine function. While the cosine function gives us values ranging between -1 and 1, the secant function tends to extend infinitely except when cosine equals zero. This means the secant function will have undefined values (gaps or vertical asymptotes) where the cosine of \( x \) is zero. This happens at values like \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Just like its cousin sinusoidal functions, the secant graph appears as a series of recurring patterns, or curves, but with notable vertical breaks.
Vertical Asymptotes
Vertical asymptotes occur in the graph of the secant function where the cosine function is zero, which means the secant function is undefined. For the basic secant function \( \sec x \), these vertical asymptotes are placed at \( x = \frac{\pi}{2} + k\pi \). When graphing a transformed secant function, such as \( y = -\frac{3}{2} \sec \pi x \), the location of vertical asymptotes is modified by the transformations applied. Here, the horizontal compression discussed later affects these positions, but the essential concept remains.
  • The asymptotes guide the placement of the U-shaped curves.
  • Between any two asymptotes exists one complete curve cycle of the secant graph.
Reflection and Stretching
Reflection and stretching are transformations applied to modify the basic appearance of a graph. Reflection flips the graph over a specified axis, whereas stretching alters the extent or size of the graph.For our function \( y = -\frac{3}{2} \sec \pi x \):
  • The negative sign indicates a reflection over the x-axis. This means the original upward U-shaped curves face downward after reflection.
  • The coefficient \( -\frac{3}{2} \) implies a vertical stretch by a factor of \( \frac{3}{2} \). Thus, where a standard curve maxed out at \( y = 1 \), it will now reach \( y = -\frac{3}{2} \).
Horizontal Compression
Horizontal compression results in squeezing the wave of a function along the x-axis. **For the function \( y = -\frac{3}{2} \sec \pi x \):**When a function has a coefficient multiplying \( x \) within the trigonometric function, it impacts the period of the curve.
  • The given coefficient \( \pi \) compresses the graph by a factor of \( \frac{1}{\pi} \).
  • This adjusts where the regular vertical asymptotes and peaks of the secant function occur.
  • Such transformations are crucial to consider in correctly plotting the graph, as they represent a change in cycle length compared to the base secant graph.

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

Write the equation for a cosecant function satisfying the given conditions. $$\text { period: } 3 \pi ; \text { range: }(-\infty,-2] \cup[2, \infty)$$

Use words (not an equation) to describe one of the quotient identities.

Exercises \(117-119\) will help you prepare for the material covered in the next section. In each exercise, complete the table of coordinates. Do not use a calculator. $$y=4 \sin \left(2 x-\frac{2 \pi}{3}\right)$$ $$\begin{array}{|c|c|c|c|c|c|} \hline \boldsymbol{X} & \frac{\pi}{3} & \frac{7 \pi}{12} & \frac{5 \pi}{6} & \frac{13 \pi}{12} & \frac{4 \pi}{3} \\ \hline \boldsymbol{y} & & & & & \\ \hline \end{array}$$

Define the sine of \(t\).

Solve: \(|2 x-3|=7\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation

Pure 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.