/*! 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 15 Determine amplitude, period, and... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 15

Determine amplitude, period, and phase shift for each function. $$ y=-2 \cos \left(2 x+\frac{\pi}{2}\right)-1 $$

Short Answer

Expert verified
Amplitude is 2, period is \(\pi\), and phase shift is \(-\frac{\pi}{4}\).

Step by step solution

01

Identify the general form of the cosine function

The general form of the cosine function is given by \[ y = A \cos(Bx + C) + D \]. Compare this general form with the given function \[ y = -2 \cos \left(2x + \frac{\pi}{2}\right) - 1 \].
02

Determine the amplitude

The amplitude is given by the absolute value of the coefficient in front of the cosine function. Here, the coefficient is -2. Thus, the amplitude is \[ \|A\| = |-2| = 2 \].
03

Determine the period

The period of the cosine function is determined by the formula \[ \frac{2\pi}{B} \]. In this case, \[ B = 2 \], so the period is \[ \frac{2\pi}{2} = \pi \].
04

Determine the phase shift

The phase shift is calculated using the formula \[ -\frac{C}{B} \]. In the given function, \[ C = \frac{\pi}{2} \] and \[ B = 2 \], so the phase shift is \[ -\frac{\frac{\pi}{2}}{2} = -\frac{\pi}{4} \].
05

Determine the vertical shift

The vertical shift is given by \[ D \]. In the given function, \[ D = -1 \], so there is a downward shift of 1 unit.

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 trigonometric function that describes the relationship between the angle of a right triangle and the length of the adjacent side to the hypotenuse. The general form of the cosine function is given by \( y = A \, \text{cos}(Bx + C) + D \). In this equation:
  • \( A \): Amplitude, representing the peak value from the midline.
  • \( B \): Affects the period or the length of one complete cycle.
  • \( C \): Phase shift, moving the graph horizontally.
  • \( D \): Vertical shift, moving the graph up or down.
Understanding each of these components is crucial for graphing and interpreting cosine functions.
amplitude calculation
The amplitude of a cosine function determines the maximum and minimum values the function can achieve relative to its midline. It is calculated using the absolute value of the coefficient \( A \) in front of the cosine. For the given function \( y = -2 \, \text{cos}(2x + \frac{\text{蟺}}{2}) - 1 \), the amplitude is found by taking \( | -2 | = 2 \). Hence, the amplitude is 2. This means the highest point of the graph will be 2 units away from the midline, and the lowest point will also be 2 units away in the opposite direction.
period calculation
The period of a cosine function determines how long it takes for one complete cycle of the wave. It is calculated using the formula \( \frac{2 \text{蟺}}{B} \). For the function \( y = -2 \, \text{cos}(2x + \frac{\text{蟺}}{2}) - 1 \), \( B = 2 \). Plugging this into the formula, we get the period: \( \frac{2 \text{蟺}}{2} = \text{蟺} \). This means the wave will repeat itself every \( \text{蟺} \) units along the x-axis.
phase shift calculation
The phase shift of a cosine function describes how much the graph is shifted horizontally from its usual position. It is given by the formula \( -\frac{C}{B} \). In our function \( y = -2 \, \text{cos}(2x + \frac{\text{蟺}}{2}) - 1 \), \( C = \frac{\text{蟺}}{2} \) and \( B = 2 \). Substituting these values, we get the phase shift: \( -\frac{\frac{\text{蟺}}{2}}{2} = -\frac{\text{蟺}}{4} \). This tells us that the graph of the cosine function is shifted to the left by \( \frac{\text{蟺}}{4} \) units.
vertical shift
The vertical shift of a cosine function indicates how much the entire graph is moved up or down along the y-axis. It is represented by the constant \( D \). For our function \( y = -2 \, \text{cos}(2x + \frac{\text{蟺}}{2}) - 1 \), the vertical shift is \( -1 \). This means the graph is shifted down by 1 unit.
To summarize:
  • Amplitude: 2
  • Period: \( \text{蟺} \)
  • Phase Shift: \( -\frac{\text{蟺}}{4} \)
  • Vertical Shift: Down 1 unit
Understanding these concepts will help you fully analyze and graph any cosine function!

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

Determine the amplitude, period, phase shift, and range for each function. $ y=-2 \cos x $$

WRITING/DISCUSSION Even or Odd An even function is one for which \(f(-x)=f(x)\) and an odd function is one for which \(f(-x)=-f(x)\). Determine whether \(f(x)=\sin (x)\) and \(f(x)=\cos (x)\) are even or odd functions and explain your answers.

Find \(B, C,\) and \(D\) so that the graph of \(y=-3 \sin [B(x-C)]+D\) is the same as the graph of \(y=3 \sin [4(x-\pi / 2)]+5\)

A weight hanging on a vertical spring is set in motion with an upward velocity of \(4 \mathrm{~cm} / \mathrm{sec}\) from its equilibrium position. A formula that gives the location of the weight in centimeters as a function of the time \(t\) in seconds is \(x=-\frac{4}{\pi} \sin (\pi t) .\) Find the period of the function and sketch its graph for \(t\) in the interval \([0,4]\) .

If \(\beta\) is an angle in standard position such that \(\sin (\beta)=1 / 4\) and \(\beta\) terminates in quadrant II, then what is the exact value of \(\cos (\beta) ?\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.