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Chapter 11: Problem 53

Without graphing, determine the amplitude and period of \(y=4 \cos \left(\frac{1}{5} x\right)\)

Short Answer

Expert verified
Amplitude: 4, Period: 10\pi

Step by step solution

01

Identify the general form of the cosine function

The given function is the cosine function which can be written in the form of \[ y = A \cos(Bx + C) + D \]Here, \(A\) is the amplitude and \(B\) is used to find the period.
02

Determine the amplitude

The amplitude \(A\) is the coefficient in front of the cosine function. For \[ y = 4 \cos(\frac{1}{5} x) \], the coefficient is 4, so the amplitude is 4.
03

Determine the period

The period of a cosine function is given by the formula \[ \text{Period} = \frac{2\pi}{B} \]. In this case, \(B\) is \(\frac{1}{5}\). Therefore, \[ \text{Period} = \frac{2\pi}{\frac{1}{5}} = 2\pi \cdot 5 = 10\pi. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

amplitude
In trigonometric functions, the amplitude is a measure of the peaks and troughs of a wave. It represents the height from the middle (or axis) of the wave to the maximum or minimum point.
In the general form of a cosine function, \( y = A \cos(Bx + C) + D \), the amplitude is denoted by the coefficient \( A \).
For the problem, given \( y = 4 \cos(\frac{1}{5} x) \), the amplitude is the coefficient before the cosine function, which is \( 4 \).
Hence, the highest point (peak) of the wave is \( 4 \) units above the midline, and the lowest point (trough) is \( 4 \) units below the midline.
This indicates how steep or flat the wave is.
period
The period of a trigonometric function is the length of one complete cycle of the wave.
For cosine functions, the period is determined by the coefficient \( B \) in the expression \( y = A \cos(Bx + C) + D \).
The formula for the period is \[ \text{Period} = \frac{2\pi}{B} \].
In the given equation \[ y = 4 \cos(\frac{1}{5} x) \], \( B = \frac{1}{5} \).
Plug \( B \) into the formula to get the period:
\ \text{Period} = \frac{2\pi}{\frac{1}{5}} = 2\pi \times 5 = 10\pi \.
This means the function repeats itself every \( 10\pi \) units.
cosine function
The cosine function is one of the fundamental trigonometric functions.
Its general form is \[ y = A \cos(Bx + C) + D \]. Here's a breakdown:
\ul>
  • \( A \): Amplitude, the height of the wave.
    • \li> \( B \): Coefficient that affects the period. \li> \( C \): Horizontal shift, which moves the wave left or right. \li> \( D \): Vertical shift, which moves the wave up or down.
    The cosine function starts at its maximum value (if the amplitude is positive) and oscillates symmetrically around its midline.
    For the function \[ y = 4 \cos(\frac{1}{5} x) \], it has:
    \ul>
  • Amplitude \ 4.
    • \li> Period \ 10\pi.
    The function describes how far and how frequently the wave oscillates.

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