Problem 10 - Calculus Concepts

Question

For each of the functions, state the amplitude, period, average value, and horizontal shift. \(\quad g(x)=\sin (x-\pi)\)

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(2\pi\), Average value: 0, Horizontal shift: \(\pi\) units right.

Step 1: Identify the Standard Form

The given function is \( g(x) = \sin(x - \pi) \). The standard form of a sine function is \( a \sin(bx - c) + d \). In this case, by comparing, we have \( a = 1 \), \( b = 1 \), and \( c = \pi \). The vertical shift \( d = 0 \).

Step 2: Determine the Amplitude

The amplitude of a sine function in the standard form is the coefficient \( a \). Here, \( a = 1 \). So, the amplitude is 1.

Step 3: Calculate the Period

The period of a sine function is found using the formula \( \dfrac{2\pi}{b} \). Given \( b = 1 \), the period is \( \dfrac{2\pi}{1} = 2\pi \).

Step 4: Find the Average Value

The average value of a basic sine function without vertical shift \( (d = 0) \) is 0. Since \( d = 0 \), the average value is 0.

Step 5: Determine the Horizontal Shift

The horizontal shift is calculated by \( \dfrac{c}{b} \). Here, \( c = \pi \) and \( b = 1 \). Thus, the horizontal shift is \( \dfrac{\pi}{1} = \pi \). Because it is \( x - \pi \), it shifts to the right by \( \pi \) units.

Key Concepts

Amplitude

In the context of trigonometric functions, amplitude reflects how far the wave of the function stretches above and below its central axis. It's an essential feature that determines the vertical height from the middle line in a sine or cosine function.
For the function \( g(x) = \sin(x - \pi) \), the amplitude can be easily identified.

• The standard form is \( a \sin(bx - c) + d \).
• The amplitude is represented by \( a \), which denotes the height of the wave from the horizontal axis.
• In this function, \( a = 1 \), meaning the wave reaches a maximum of 1 and a minimum of -1 from the middle line.

Thus, the amplitude of \( g(x) = \sin(x - \pi) \) is 1, indicating that the wave maintains a consistent motion of one unit both upwards and downwards from its central line.

Period

The period of a trigonometric function describes how long it takes for the function to complete one full cycle of its pattern. It's the horizontal length required for the function to repeat itself.

• For sine functions, the period is determined by the formula \( \dfrac{2\pi}{b} \), where \( b \) is the coefficient of \( x \).
• The function \( g(x) = \sin(x - \pi) \) has \( b = 1 \).
• Substituting \( b = 1 \) into the period formula gives \( \dfrac{2\pi}{1} = 2\pi \).

This tells us that the function repeats its pattern every \( 2\pi \) units along the horizontal axis, providing the fundamental frequency of the function.

Horizontal Shift

Horizontal shift involves shifting the entire graph of the function left or right along the x-axis. It's determined by the phase shift in trigonometric functions.

• Given the function \( g(x) = \sin(x - \pi) \), the horizontal shift is given by \( \dfrac{c}{b} \).
• Here, \( c = \pi \) and \( b = 1 \).
• Thus, \( \dfrac{\pi}{1} = \pi \) shows the function shifts to the right by \( \pi \) units.
• Since the shift is represented within the sine function, \( x - \pi \) indicates a movement in the positive x-direction.

This adjustment leads to the entire sine curve being realigned rightward without altering its shape or amplitude.

Average Value

The average value of sine and cosine functions identifies the function’s mean position over one cycle. It's crucial because it helps in understanding the baseline around which the function oscillates.
For standard sine functions like \( g(x) = \sin(x - \pi) \), the key points include:

• The average value is determined by the vertical shift \( d \) in the function's form \( a \sin(bx - c) + d \).
• In this function, the vertical shift \( d = 0 \), meaning there's no displacement from the x-axis.
• Therefore, the average value remains constant at 0 for the function, corresponding neatly with its central axis.

This baseline of 0 is pivotal as it shows that the oscillations of the wave maintain balance around the horizontal axis, averaging out at zero over its cycle.