Problem 169 - Calculus Volume 1

Question

Find a. the amplitude, b. the period, and c. the phase shift with direction for each function. \(y=\frac{-1}{2} \sin \left(\frac{1}{4} x\right)\)

Step-by-Step Solution

Answer

Amplitude: \(\frac{1}{2}\); Period: \(8\pi\); Phase Shift: None.

Step 1: Identify the General Form of the Function

The general form of a sine function is \( y = a \sin(bx + c) + d \). For the function \( y = \frac{-1}{2} \sin \left(\frac{1}{4} x\right) \), identify \(a\), \(b\), and \(c\) by comparing it with the general form. Here, \(a = \frac{-1}{2}\), \(b = \frac{1}{4}\), and \(c = 0\).

Step 2: Find the Amplitude

The amplitude of a sine function is the absolute value of \(a\). Given \(a = \frac{-1}{2}\), the amplitude is \(\left|\frac{-1}{2}\right| = \frac{1}{2}\).

Step 3: Calculate the Period

The period of a sine function is given by \(\frac{2\pi}{b}\). With \(b = \frac{1}{4}\), compute the period as \(\frac{2\pi}{\frac{1}{4}} = 8\pi\).

Step 4: Determine the Phase Shift

The phase shift of a sine function is calculated by \(-\frac{c}{b}\). Since \(c = 0\), the phase shift is \(-\frac{0}{\frac{1}{4}} = 0\). There is no phase shift in this function.

Key Concepts

Amplitude

In trigonometric functions, the amplitude refers to the measure of how much a wave varies in the vertical direction. Simply put, it's the height from the centerline of the wave to its peak or trough. In any function of the form \( y = a \sin(bx + c) + d \), the amplitude is determined by the coefficient \( a \).
For our specific function, \( y = \frac{-1}{2} \sin \left(\frac{1}{4} x\right) \), the amplitude is the absolute value of \( a \), which is \( \left| \frac{-1}{2} \right| \).
Consequently, the amplitude is \( \frac{1}{2} \). This means the sine wave reaches a maximum height of \( \frac{1}{2} \) units above and below the centerline. The amplitude remains unaffected by changes in the frequency or the horizontal position of the function.

Period of a Function

The period of a sine or cosine function is a measure of the horizontal length before the function starts repeating itself. For the standard trigonometric function \( y = a \sin(bx + c) + d \), the period is calculated by the formula \( \frac{2\pi}{b} \).
In the given function, \( y = \frac{-1}{2} \sin \left(\frac{1}{4} x\right) \), the coefficient \( b \) is \( \frac{1}{4} \). Therefore, the period is \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).
This indicates that it takes \( 8\pi \) units along the x-axis for the sine wave to complete one full cycle and begin to repeat its pattern.

• A smaller value of \( b \) stretches the period to a larger number, making the wave repeat less frequently.
• A larger value of \( b \) results in a shorter period, causing the wave to cycle more quickly.

Phase Shift

Phase shift refers to the horizontal movement of a trigonometric wave along the x-axis. It indicates how much the wave is shifted left or right from the usual position without moving vertically or changing its size. In the generic formula \( y = a \sin(bx + c) + d \), the phase shift is calculated using \( -\frac{c}{b} \).
For the function \( y = \frac{-1}{2} \sin \left(\frac{1}{4} x\right) \), since \( c \) is zero, the phase shift is \( -\frac{0}{\frac{1}{4}} = 0 \). Therefore, there is no horizontal shift in this function. The wave starts right at the origin without any delay or advance.

• If \( c \) were positive, it would shift the wave to the left.
• If \( c \) were negative, it would shift the wave to the right.

This helps you understand how trigonometric functions can move along the graph's horizontal axis without altering their amplitude or period.