Problem 169 - Calculus Volume 1

Question

For the following exercises, 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 (0).

Step 1: Identify the Form of the Sine Function

The given function is \( y = \frac{-1}{2} \sin \left(\frac{1}{4} x\right) \). A general sine function can be represented as \( y = a \sin(bx + c) + d \), where \( a \) is the amplitude, \( b \) affects the period, and \( c \) is the phase shift. In our function, \( a = \frac{-1}{2} \), \( b = \frac{1}{4} \), and the phase shift component \( c = 0 \).

Step 2: Calculate the Amplitude

The amplitude \( A \) of the sine function is the absolute value of \( a \). Therefore, the amplitude is given by \( |\frac{-1}{2}| = \frac{1}{2} \).

Step 3: Determine the Period

The period \( T \) of a sine function is calculated using the formula \( T = \frac{2\pi}{|b|} \). For our function, \( b = \frac{1}{4} \). Substitute this value into the formula to get \( T = \frac{2\pi}{\frac{1}{4}} = 8\pi \).

Step 4: Find the Phase Shift

The phase shift \( \phi \) is calculated from the expression \( bx + c \) in the format \( \sin(bx + c) \). Here, it is \( \sin(\frac{1}{4}x) \). Since \( c = 0 \), the phase shift is zero, meaning there is no horizontal shift. The direction is neither left nor right since there is no shift.

Key Concepts

Amplitude

Amplitude refers to the height of the wave from the center line to the peak or trough of the wave in a trigonometric function. In simple terms, it determines how tall or short the wave appears. For a sine function expressed in the form \( y = a \sin(bx+c) + d \), the amplitude is represented by the absolute value of \( a \).

This means amplitude is always a positive quantity as it denotes a distance. In our specific case with the function \( y = \frac{-1}{2} \sin \left(\frac{1}{4} x \right) \), the amplitude is \( |\frac{-1}{2}| = \frac{1}{2} \).

Remember:

• Amplitude affects the vertical stretch or compression of the graph.
• If \( a \) is negative, it flips the graph over the horizontal axis, but the amplitude remains positive.

So here, the negative sign would cause the sine wave to invert, but the height of that wave remains \( \frac{1}{2} \) from the center line.

Period

The period of a function defines the interval it takes for the function to complete one full cycle. For trigonometric functions, this is essentially how "stretched" or "compressed" the wave appears horizontally.

The basic formula to determine the period of a sine or cosine wave is \( T = \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \) in \( y = a \sin(bx+c) + d \).

In our case, with \( b = \frac{1}{4} \), we calculate:\[T = \frac{2\pi}{|\frac{1}{4}|} = 8\pi\]This result, \( 8\pi \), tells us that it takes a distance of \( 8\pi \) units along the x-axis for the wave to complete one full cycle. This is considerably longer than the default period of a standard sine function \( y = \sin x \), which is \( 2\pi \). Always keep in mind:

• The period indicates how frequent the cycles of the wave are.
• The smaller the \( b \), the larger the period, and vice versa.

Phase Shift

Phase shift refers to the horizontal displacement of the function along the x-axis. It allows you to determine if the wave starts later or earlier than the standard position.

For a sine function given by \( y = a \sin(bx+c) + d \), the phase shift can be calculated from \( cx \) in the expression \( bx + c \). It is defined as \( -\frac{c}{b} \).

In our function \( y = \frac{-1}{2} \sin \left(\frac{1}{4} x \right) \), we have \( c = 0 \). This simplifies the calculation to:\[\text{Phase shift} = -\frac{0}{\frac{1}{4}} = 0\]This indicates there is no phase shift. The graph starts in the usual position neither shifted to the left nor the right.

Points to note about phase shift:

• A positive phase shift moves the function left; a negative shift moves it right.
• No value for \( c \) means no phase shift occurred.