/*! 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 22 In Exercises 9-24, sketch the gr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 22

In Exercises 9-24, sketch the graph of each sinusoidal function over one period. $$ y=-2+\cos \left(\frac{\pi}{2} x\right) $$

Short Answer

Expert verified
The period of the function is 4 and it is a shifted cosine wave down by 2 units.

Step by step solution

01

Understand the Components of the Function

The given function is \( y = -2 + \cos \left( \frac{\pi}{2} x \right) \). This is a transformation of the basic cosine function \( y = \cos(x) \). Let's identify the transformations: **Vertical Shift:** Down 2 units (constant term "-2"). **Horizontal Compression:** The coefficient \( \frac{\pi}{2} \) modifies the period.
02

Calculate the Period of the Function

The period of a cosine function is \( 2\pi \). However, when a function is of the form \( \cos(bx) \), the period is \( \frac{2\pi}{b} \). In this case, \( b = \frac{\pi}{2} \), so the new period is: \[ \text{Period} = \frac{2\pi}{\frac{\pi}{2}} = 4 \]
03

Sketch the Graph over One Period

1. **Identify Key Points:** Divide the period (4) into 4 equal parts to determine key points: 0, 1, 2, 3, and 4. 2. **Start Sketching:** At \( x = 0 \), \( y = -2 + \cos(0) = -1 \). 3. **Continue Over the Period:** - At \( x = 1 \), \( y = -2 + \cos\left( \frac{\pi}{2} \right) = -2 \). - At \( x = 2 \), \( y = -2 + \cos(\pi) = -3 \). - At \( x = 3 \), \( y = -2 + \cos\left( \frac{3\pi}{2} \right) = -2 \). - At \( x = 4 \), \( y = -2 + \cos(2\pi) = -1 \). 4. **Plot the Points and Connect:** Plot these points on a graph with the x-axis scale from 0 to 4 and y-axis based on values calculated. Then smoothly connect them following the cosine wave pattern.

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.

Cosine Function
The cosine function is a fundamental aspect of trigonometry, often written as \( y = \cos(x) \). The graph of this function highlights its periodic nature, resembling a wave that oscillates smoothly between peaks and valleys.
  • The cosine function starts at its maximum value, decreasing to a minimum before reaching the maximum again, completing one cycle.
  • Its maximum value is 1 and minimum value is -1 when unaltered.
  • First, it peaks, then crosses the midline, dips to the lowest point, and returns to the midline.
The classic cosine wave is often the basis for applying transformations, as seen in our function \( y = -2 + \cos \left( \frac{\pi}{2} x \right) \). Understanding these characteristics aids in predicting and sketching cosine-related graphs.
Periodic Functions
A key attribute of the cosine function, like other trigonometric functions, is that it is periodic. This means its pattern repeats at regular intervals.
  • The standard period of a cosine function is \( 2\pi \), signifying the length required for one full cycle.
  • Modifying the period changes the frequency of oscillations across a given domain.
In the exercise, the period was adjusted by a horizontal compression factor \( \frac{\pi}{2} \). Applying this to \( \cos \left( \frac{\pi}{2} x \right) \), we calculated the new period as \( 4 \) instead of the original \( 2\pi \). Recognizing how to find and adjust the period is vital in graph manipulation and interpretation.
Graph Transformations
Graph transformations are essential to altering the basic sine or cosine wave through shifts, stretches, or compressions. They help in adapting a graph to fit certain patterns or data.
  • Transformations can move the graph vertically or horizontally, adjust its width, or change its amplitude.
  • Understanding each type of transformation individually aids in predicting their combined effect on the graph.
For \( y = -2 + \cos \left( \frac{\pi}{2} x \right) \), we identified both vertical shift and horizontal compression as transformations. Recognizing these transformations makes it easier to sketch accurate graphs that reflect these modifications.
Vertical Shift
A vertical shift occurs when the entire graph moves up or down along the y-axis. In the context of our function, the term \(-2\) signifies a downward shift.
  • This means each point on the cosine graph moves down two units.
  • The midline, originally at \( y = 0 \), shifts to \( y = -2 \).
  • This shift does not affect the period or frequency but moves the waveform lower on the graph.
Being able to visualize and apply vertical shifts in trigonometric graph transformations enhances graph accuracy and aids understanding of function behavior.
Horizontal Compression
Horizontal compression occurs when a graph is squeezed along the x-axis. This changes the frequency of the function's waves.
  • For \( \cos \left( \frac{\pi}{2} x \right) \), the compression is due to the factor \( \frac{\pi}{2} \).
  • This leads the cosine wave to complete its pattern more quickly, reducing the period to 4.
  • Squeezing results in more cycles over the same interval, increasing the apparent frequency.
Understanding horizontal compression allows one to adjust the speed at which a trigonometric function completes its cycles, a critical skill for analyzing waveforms of all types.

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

In Exercises 51-56, state the domain and range of the functions. $$ y=1-2 \sec \left(\frac{1}{2} x+\pi\right) $$

In Exercises 19-30, graph the functions over the indicated intervals. $$ y=2 \tan \left(\frac{1}{3} x\right),-3 \pi \leq x \leq 3 \pi $$

For Exercises 65-68, refer to the following: Computer sales are generally subject to seasonal fluctuations. An analysis of the sales of QualComp computers during 2008-2010 is approximated by the function $$ s(t)=0.098 \sin (0.79 t+2.37)+0.387 \quad 1 \leq t \leq 12 $$ where \(t\) represents time in quarters ( \(t=1\) represents the end of the first quarter of 2008), and s(t) represents computer sales (quarterly revenue) in millions of dollars. $$ \text { Business/Economics. Find the period. Interpret its meaning. } $$

For Exercises 71 and 72 , find a sinusoidal function of the form \(y=k+A \sin (B x+C)\) that fits the data, where \(y\) is the temperature in degrees Fahrenheit and \(x\) is the number of the month with \(x=1\) corresponding to January. Assume the period is 12 months. Temperature. The average temperature in Houston, Texas, is \(67.9^{\circ} \mathrm{F}\). The average monthly temperatures are given by Average Temperature by Month \(\left({ }^{\circ} \mathrm{F}\right)\) Houston, TX Jan Feb Mar Apr May Jun Jul Aug Sep Oet Nov Dec \(\begin{array}{llllllllllll}50.4 & 53.9 & 60.6 & 68.3 & 74.5 & 80.4 & 82.6 & 82.3 & 78.2 & 69.6 & 61.0 & 53.5\end{array}\)

$$ \text { Find the domain for the function } y=\cot (4 x-\pi) \text {. } $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

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.