Problem 12 - Calculus

Question

Find the period and amplitude. $$w=8-4 \sin (2 x+\pi)$$

Step-by-Step Solution

Answer

Amplitude is 4 and period is \(\pi\).

Step 1: Understand the Sine Function

The general form of a sine function is \(y = A \sin(Bx + C) + D\), where: \(A\) is the amplitude, \(B\) affects the period, \(C\) is the phase shift, and \(D\) translates the graph vertically.

Step 2: Identify Parameters in the Given Function

The given function is \(w = 8 - 4 \sin(2x + \pi)\). We can rewrite this as \(w = -4 \sin(2x + \pi) + 8\), identifying \(A = -4\), \(B = 2\), \(C = \pi\), and \(D = 8\).

Step 3: Calculate the Amplitude

The amplitude is the absolute value of \(A\). Thus, amplitude = \(|-4| = 4\).

Step 4: Calculate the Period

The period of a sine function is given by \(\frac{2\pi}{B}\). Since \(B = 2\), the period is \(\frac{2\pi}{2} = \pi\).

Key Concepts

Amplitude

When exploring trigonometric functions, "amplitude" is a crucial concept. Amplitude describes the height of the wave created by the sine function. In simpler terms, it is the distance from the midline of the wave to its peak or trough.

For the sine function of the form \( y = A \sin(Bx + C) + D \), the amplitude is given by the absolute value of \( A \). This ensures that the amplitude is always a positive number, representing a distance. In our provided example, the function is \( w = 8 - 4 \sin(2x + \pi) \). After rearranging, we identify that \( A = -4 \).

• The amplitude is therefore \( \left|-4\right| = 4 \).

This tells us that the total vertical distance between the highest and lowest points of the sine wave is 8 units, since the wave reaches 4 units above and below its central value.

Period

The concept of "period" in a sine function relates to how long it takes for the function to complete one full cycle. In mathematical terms, the period is the horizontal length over which the function repeats itself. This is crucial for understanding how frequently waves generated by trigonometric functions oscillate.

For a sine function represented as \( y = A \sin(Bx + C) + D \), the formula for determining the period is \( \frac{2\pi}{B} \). The value of \( B \) affects how "stretched" or "compressed" the wave appears along the x-axis. In our function \( w = 8 - 4 \sin(2x + \pi) \), \( B \) is equal to 2.

• Thus, the period is \( \frac{2\pi}{2} = \pi \).

This means that every \( \pi \) units along the x-axis, the sine wave will repeat its behavior, maintaining a consistent oscillation.

Sine Function

The sine function is one of the fundamental functions in trigonometry and is defined by its characteristic wave-like pattern. Trigonometric functions such as sine are essential in various fields including physics, engineering, and music. They represent periodic phenomena like sound waves, light waves, and the oscillation of pendulums.

The sine function specifically is expressed in the formula \( y = A \sin(Bx + C) + D \), where each constant has a specific role:

• \( A \) determines the amplitude, affecting the height of the wave.
• \( B \) influences the period, impacting how frequently the wave repeats.
• \( C \) handles the phase shift, moving the wave left or right.
• \( D \) adjusts the vertical translation, moving the entire wave up or down.

Understanding these elements helps in dissecting complex wave patterns and applying them effectively to real-world scenarios, as seen in our exercise with the altered sine function \( w = 8 - 4 \sin(2x + \pi) \). Here, dealing with each parameter reveals the function's amplitude, period, and vertical and horizontal shifts.