/*! 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 38 Determine the amplitude, period,... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 38

Determine the amplitude, period, and phase shift for each function. $$y=-2 \cos (3 x)$$

Short Answer

Expert verified
Amplitude: 2, Period: \( \frac{2\pi}{3} \), Phase Shift: 0

Step by step solution

01

Identify the form of the function

The given function is of the form: \[ y = a \, \text{cos}(b x + c) \] Here, \( a = -2 \), \( b = 3 \), and \( c = 0 \).
02

Find the amplitude

The amplitude of the function is given by the absolute value of \(a\). So, amplitude = \( |-2| = 2 \).
03

Determine the period

The period of a function \( y = a \, \text{cos}(b x + c) \) is given by the formula: \[ \text{Period} = \frac{2\pi}{|b|} \] Substitute \( b = 3 \): \[ \text{Period} = \frac{2\pi}{3} \]
04

Calculate the phase shift

The phase shift is given by the formula: \[ \text{Phase Shift} = -\frac{c}{b} \] Since \( c = 0 \), \[ \text{Phase Shift} = -\frac{0}{3} = 0 \]

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.

Amplitude
The amplitude of a trigonometric function represents the height of the wave from the midline to its peak. For the given function, we have:

For example, in the function \[ y = -2 \text{cos}(3x) \]
The amplitude is found by taking the absolute value of the coefficient in front of the cosine function. This coefficient is referred to as 'a'. Thus, we have:
  • \[ a = -2 \]
  • Amplitude = \[ |-2| = 2 \]
The amplitude of 2 indicates that the wave reaches 2 units above and below its midline. When dealing with negative coefficients like -2, the negative sign signifies a reflection over the horizontal axis.
Period
The period of a trigonometric function is the distance over which the function repeats itself. For the given cosine function
\[ y = -2 \text{cos}(3x) \]
The period is calculated using the formula:
  • Period = \[ \frac{2\pi}{|b|} \]
Here, 'b' is the coefficient of 'x' inside the cosine function. For our function:
  • \[ b = 3\]
Substitute this value into the formula to find the period:
  • Period = \[ \frac{2\pi}{3} \]
This means that the wave will repeat after every \[ \frac{2\pi}{3} \] units along the x-axis. A smaller value of 'b' would result in a longer period, and vice-versa.
Phase Shift
Phase shift in a trigonometric function indicates the horizontal shifting of the function along the x-axis. It's determined by the constant 'c' inside the expression. For our function:
\[ y = -2 \text{cos}(3x) \]
The phase shift is calculated using the formula:
  • Phase Shift = \[ -\frac{c}{b} \]
In this specific function, 'c' is zero:
  • \[ c = 0 \]
  • \[ b = 3 \]
  • Phase Shift = \[ -\frac{0}{3} = 0 \]
Since 'c' is zero, the function has no horizontal shift and starts at the origin. If 'c' were a non-zero value, the function would shift either left or right depending on the sign of 'c'.

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

Solve each problem. Photographing the Earth From an altitude of \(161 \mathrm{mi}\), the Orbiting Geophysical Observatory (OGO-1) can photograph a path on the surface of the earth that is approximately \(2000 \mathrm{mi}\) wide. Find the central angle, to the nearest tenth of a degree, that intercepts an arc of \(2000 \mathrm{mi}\) on the surface of the earth (radius 3950 mi). (IMAGE CANNOT COPY)

Determine the period and sketch at least one cycle of the graph of each function. State the range of each function. $$y=\sec (3 x)$$

Area of a Slice of Pizza If a 16 -in.-diameter pizza is cut into six slices of the same size, then what is the area of each piece?

Find the exact value of each expression for the given value of \(\theta .\) Do not use a calculator. $$\sec ^{2}(2 \theta) \text { if } \theta=\pi / 6$$

Find the approximate value of each expression. Round to four decimal places. $$\tan \left(-44.6^{\circ}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.