Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=6 \sin \pi x $$

Step-by-Step Solution

Answer

Amplitude: 6, Period: 2, Phase Shift: 0. Graph is a sine wave with these parameters.

Step 1: Identify the Amplitude

The standard form of a sine function is \( y = A \sin(Bx - C) + D \). In this equation \( y = 6 \sin \pi x \), the amplitude \( A \) is the coefficient of the sine function, which is 6.

Step 2: Determine the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{B} \). In the function \( y = 6 \sin \pi x \), the value of \( B \) is \( \pi \). Thus, the period is \( \frac{2\pi}{\pi} = 2 \).

Step 3: Calculate the Phase Shift

The phase shift of the function is calculated by \( \frac{C}{B} \). The equation can be rearranged as \( y = 6 \sin(\pi x - 0) \) which shows \( C = 0 \). Therefore, the phase shift is \( \frac{0}{\pi} = 0 \), indicating no phase shift.

Step 4: Plot the Graph

The function \( y = 6 \sin \pi x \) has an amplitude of 6, a period of 2, and no phase shift. The graph starts at \( (0,0) \), reaches its maximum value at \( (0.5, 6) \), returns through the origin at \( (1,0) \), reaches its minimum value at \( (1.5, -6) \), and completes one cycle back at \( (2,0) \).

Key Concepts

Amplitude of Sine Function

When you're dealing with the sine function, understanding the concept of "amplitude" is crucial. The amplitude of a sine wave determines the height of the wave from the center line to the peak. It's essentially how "tall" or "short" the wave is. In the general form of the sine function, which is \( y = A \sin(Bx - C) + D \), the amplitude is represented by \( A \).
For example, in the equation \( y = 6 \sin \pi x \), the amplitude is \( 6 \). This means:

• The sine wave reaches a maximum height of 6 above the center line.
• It also dips to a minimum of 6 below the center line, which gives a total "height" of 12 from peak to trough.

This amplitude value dictates how much the function moves vertically.
Without affecting the period or shift horizontally, changing the amplitude will simply stretch or compress the sine wave vertically.

Period of Sine Function

The "period" of a sine function defines how long it takes for the function to complete one full cycle of its wave. To find the period of a sine function given in the form \( y = A \sin(Bx - C) + D \), use the formula for the period which is \( \frac{2\pi}{B} \).
For the function \( y = 6 \sin \pi x \), let's find the period:

• Here, \( B \) is given as \( \pi \).
• The period is thus \( \frac{2\pi}{\pi} = 2 \).

This means that the function repeats itself every two units.
In a general sense, a smaller \( B \) value results in a longer period, stretching the wave horizontally, while a larger \( B \) value shortens the period, compressing the wave horizontally. This is a key feature to understand when graphing trigonometric functions.

Phase Shift in Trigonometry

Phase shift refers to the horizontal movement of the sine wave along the x-axis. It's how the whole wave shifts left or right from its original position. To determine the phase shift of a function given as \( y = A \sin(Bx - C) + D \), you can use the formula \( \frac{C}{B} \).
In the equation \( y = 6 \sin(\pi x) \), let's find the phase shift:

• The rearranged function can be seen as \( \pi x - 0 \), indicating \( C = 0 \).
• Hence, the phase shift is \( \frac{0}{\pi} = 0 \).

This means there is no horizontal shift in this particular function. It starts exactly at the origin where a standard sine function would start.
If \( C \) were not zero, the phase shift would be influenced by \( C \) displaced according to \( B \). A positive \( C \) would shift the graph to the right, while a negative \( C \) would shift it to the left. Understanding phase shifts helps in perfectly aligning the sine or cosine function to match any external conditions or phenomena.