/*! 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 42 Graph each function using transf... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 42

Graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=2 \cos \left(\frac{1}{4} x\right) $$

Short Answer

Expert verified
The domain is \( -\infty, \infty \) and the range is \( [-2, 2] \).

Step by step solution

01

Identify the Parent Function

The parent function is the cosine function, which is generally written as \( y = \cos(x) \).
02

Determine Amplitude and Vertical Shift

The coefficient 2 in front of the cosine function indicates that the amplitude is 2. There's no vertical shift because there's no constant added or subtracted from the function.
03

Determine the Period

The coefficient \( \frac{1}{4} \) affects the period of the function. The period of a cosine function is given by \( 2\pi \div \frac{1}{4} = 8\pi \).
04

Key Points of the Parent Function

For the parent function \( y = \cos(x) \), the key points within one period (0 to \( 2\pi \)) are: \[ \begin{align*} (0, 1), \ (\frac{\pi}{2}, 0), \ (\pi, -1), \ (\frac{3\pi}{2}, 0), \ (2\pi, 1) \end{align*} \]
05

Apply Transformation to Key Points

Using the scaling factor \( \frac{1}{4} \) for the x-values and multiplying the y-values by 2, the transformed key points are: \[ \begin{align*} (0, 2), \ (2\pi, 0), \ (4\pi, -2), \ (6\pi, 0), \ (8\pi, 2) \end{align*} \]
06

Plot Key Points and Draw Graph

Plot the transformed key points on a graph and draw the cosine curve through these points. Ensure that at least two cycles are represented.
07

Determine the Domain and Range

The domain of \( y = 2 \cos( \frac{1}{4} x) \) is all real numbers (\( -\infty, \infty \)). The range, given the amplitude of 2, is \( [-2, 2] \).

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.

Understanding the Cosine Function
The cosine function, denoted as \( y = \cos(x) \), is a fundamental trigonometric function. Its graph represents the relationship between an angle and the ratio of the adjacent side to the hypotenuse in a right triangle.

Key points of the cosine function include:
  • At \( x = 0 \), \( y = 1 \)
  • At \( x = \frac{\pi}{2} \), \( y = 0 \)
  • At \( x = \pi \), \( y = -1 \)
  • At \( x = \frac{3\pi}{2} \), \( y = 0 \)
  • At \( x = 2\pi \), \( y = 1 \)
The cosine function oscillates between 1 and -1, creating a periodic wave. Each full cycle, from 0 to \( 2\pi \), captures the repetitive nature of this function. This should give us the intuition required to dive deeper into trigonometric transformations.
Trigonometric Transformations
Trigonometric transformations modify the properties of trigonometric functions like the cosine function.

Let's decipher our specific function, \( y = 2 \cos \( \frac{1}{4} x \) \). This involves multiple transformations:
  • **Amplitude Change:** The coefficient 2 means the function鈥檚 peaks and troughs are scaled vertically by a factor of 2. Therefore, the maximum and minimum values are now 2 and -2 respectively, instead of 1 and -1.
  • **Period Change:** The factor \( \frac{1}{4} \) alters the period. The period of the standard cosine function is \( 2\pi \). However, it is scaled by \( \frac{1}{4} \), so the new period is calculated as \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).
How do these transformations affect our graph?
  • Take the key points of the usual cosine function.
  • Adjust the x-values by multiplying by 4 (due to the factor \( \frac{1}{4} \)).
  • Multiply the y-values by 2.
This results in new key points:
  • \( (0, 2) \)
  • \( (2\pi, 0) \)
  • \( (4\pi, -2) \)
  • \( (6\pi, 0) \)
  • \( (8\pi, 2) \)
Graphing these points will show how the function is transformed, maintaining the cosine wave's characteristic shape but with altered amplitude and period.
Domain and Range
The **domain** of a function represents all the x-values where the function is defined. For our trigonometric function \(y = 2\cos(\frac{1}{4} x)\):
  • The cosine function is defined for all x-values.
  • Thus, the domain is all real numbers: \( (-\infty, \infty) \).
The **range** of a function indicates all possible y-values it can take. Considering that the function鈥檚 amplitude is 2:
  • The cosine function has been scaled vertically by 2.
  • So the maximum value is 2, and the minimum value is -2.
  • Therefore, the range of this function is: \([-2, 2]\).
These transformations help us understand and predict the behavior of trigonometric functions, making it easier to work with them in various mathematical contexts.

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

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle \(\theta\). $$\csc \theta=2$$

Problems \(63-66\) require the following discussion. Projectile Motion The path of a projectile fired at an inclination \(\theta\) to the horizontal with initial speed \(v_{0}\) is a parabola. See the figure. The range \(R\) of the projectile-that is, the horizontal distance that the projectile travels-is found by using the function $$ R(\theta)=\frac{2 v_{0}^{2} \sin \theta \cos \theta}{g} $$ where \(g \approx 32.2\) feet per second per second \(\approx 9.8\) meters per second per second is the acceleration due to gravity. The maximum height \(H\) of the projectile is given by the function $$ H(\theta)=\frac{v_{0}^{2} \sin ^{2} \theta}{2 g} $$ Find the range \(R\) and maximum height \(H\) of the projectile. Round answers to two decimal places. The projectile is fired at an angle of \(45^{\circ}\) to the horizontal with an initial speed of 100 feet per second.

Determine the amplitude and period of each function without graphing. $$ y=\frac{4}{3} \sin \left(\frac{2}{3} x\right) $$

Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of each expression. Do not use a calculator. $$1-\cos ^{2} 15^{\circ}-\cos ^{2} 75^{\circ}$$

A point on the terminal side of an angle \(\theta\) in standard position is given. Find the exact value of each of the six trigonometric functions of \(\theta .\) $$ \left(-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2}\right) $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics 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.