/*! 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 41 Sketch the graph of the function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 41

Sketch the graph of the function. (Include two full periods.) $$ y=\frac{1}{3} \cos x $$

Short Answer

Expert verified
The graph of \( y=\frac{1}{3} \cos x \) will look like a normal cosine function but will have an amplitude of \( \frac{1}{3} \), and the period of \( 2\pi \). It will pass through the points (0, \( \frac{1}{3} \)), (\( \frac{1}{2}\pi \), 0), (\(\pi \), -\( \frac{1}{3} \)), (\( \frac{3}{2}\pi \), 0), and (\(2\pi \), \( \frac{1}{3} \)) in one cycle with no phase shift. Repeat the pattern for the second period.

Step by step solution

01

Identify the Amplitude, Period, and Phase Shift

The amplitude of the function is the absolute value of the coefficient in front of the cosine function, in this case \( \frac{1}{3} \). So, the amplitude is \( \frac{1}{3} \). The period of a basic cosine function is \( 2\pi \) and in this function, there is no coefficient with x, so period remains \( 2\pi \). The phase shift in this function is 0, as there's no addition or subtraction inside the 'cos'.
02

Plot Key Points

Plot the key points for the standard cosine function, starting from 0. Key points include: (0, amplitude), (\( \frac{1}{4} \) of period, 0), (\( \frac{1}{2} \) of period, -amplitude), (3 * \( \frac{1}{4} \) of period, 0), and (period, amplitude). For this function: (0, \( \frac{1}{3} \)), (\( \frac{1}{2}\pi \), 0), (\(\pi \), -\( \frac{1}{3} \)), (\( \frac{3}{2}\pi \), 0), and (\(2\pi \), \( \frac{1}{3} \)).
03

Sketch the Graph

Draw the function by connecting the key points plotted in the previous step, keeping in mind the wave like pattern of the cosine function. Remember to include two full periods.

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
When graphing trigonometric functions like the cosine function, one of the first things to determine is the amplitude. The amplitude describes how "tall" or "short" the wave of the graph will be. It is the absolute value of the coefficient in front of the cosine function. For the function given by y = \( \frac{1}{3} \cos x \), this coefficient is \( \frac{1}{3} \). Therefore, the amplitude of this function is \( \frac{1}{3} \).
  • This means the maximum height of the graph from the midline is \( \frac{1}{3} \).

  • Similarly, the graph will dip \( \frac{1}{3} \) units below the midline.

Calculating the amplitude is essential as it allows us to plot and understand how the graph fluctuates above and below its central axis.
Period of Trigonometric Functions
The period of a trigonometric function like cosine is the distance required for the function to complete one full cycle. For the basic cosine function y = \( \cos x \), the period is \( 2\pi \), meaning the wave-like pattern repeats every \( 2\pi \) units of x.
  • If there were a coefficient multiplying x, it would affect the period by compressing or stretching it.

  • In general, the formula for the period of a cosine function is \( \frac{2\pi}{b} \), where \(b\) is the coefficient of \(x\).

In the exercise, y = \( \frac{1}{3} \cos x \), there is no coefficient affecting \(x\) so the period remains \( 2\pi \). Understanding the period is key as it allows us to define the interval we need to graph and ensures the function replicates correctly.
Cosine Function
The cosine function is a fundamental trigonometric function that produces a wave-like graph known for its smooth, repetitive oscillations. It starts at its maximum, descends to a minimum, and returns to the maximum, completing one cycle.
To graph a cosine function like y = \( \frac{1}{3} \cos x \), one should:
  • Identify key points: The standard key points for a cosine function at the start, middle, and end of a period make sketching easier. For example, these include where it starts at maximum, crosses the midline, reaches a minimum, crosses the midline again, and returns to the maximum.

  • Consider transformations: Unlike transformations such as vertical shifts or phase shifts, this function's midline is the x-axis and doesn't have horizontal shifts.

Plot these points carefully to ensure the graph reflects the correct number of periods, oscillating both above and below the midline according to the amplitude of the 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

Use a graphing utility to graph the function. Include two full periods. $$ y=2 \sec (2 x-\pi) $$

Determine whether the statement is true or false. Justify your answer. The graph of \(y=\sec x\) can be obtained on a calculator by graphing a translation of the reciprocal of \(y=\sin x\)

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the results algebraically. $$ y_{1}=\frac{\cos x}{\sin x}, \quad y_{2}=\cot x $$

Use the graph of the function to determine whether the function is even, odd, or neither. Verify your answer algebraically. $$ g(x)=x^{2} \cot x $$

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to determine whether the expressions are equivalent. Verify the results algebraically. $$ y_{1}=\sin x \sec x, \quad y_{2}=\tan x $$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Applied 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.