Problem 17 - 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. $$ y=-2 \sin (3 x-\pi) $$

Step-by-Step Solution

Answer

Amplitude: 2, Period: \(\frac{2\pi}{3}\), Phase shift: \(\frac{\pi}{3}\) right.

Step 1: Identify the Amplitude

The amplitude of the function is determined by the coefficient of the sine function. In the equation \( y = -2 \sin(3x - \pi) \), the amplitude is the absolute value of \(-2\). Therefore, the amplitude is 2.

Step 2: Determine the Period

The period of a sine function is given by \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) inside the sine function. Here, \( B = 3 \). Thus, the period is \( \frac{2\pi}{3} \).

Step 3: Find the Phase Shift

The phase shift is determined by the expression \( \frac{C}{B} \), where \( C \) is the constant subtracted from \( Bx \). In this case, the equation \( 3x - \pi \) suggests that \( C = \pi \) and \( B = 3 \). Thus, the phase shift is \( \frac{\pi}{3} \) to the right.

Step 4: Sketch the Graph

Begin by marking the phase shift of \( \frac{\pi}{3} \) on the x-axis. Each cycle of the sine function will span \( \frac{2\pi}{3} \). Considering the amplitude of 2, the graph will oscillate between -2 and 2, and because the sine function is negative, it will start at the minimum point (-2) before completing the cycle.

Key Concepts

Amplitude

Understanding the amplitude of a sine function is crucial, as it tells us how high and low the wave reaches from its central axis. In the equation \( y = -2 \sin(3x - \pi) \), the amplitude is represented by the coefficient in front of the sine function, which is \(-2\). The amplitude is the absolute value of this coefficient, so if we take \( |-2| = 2 \), this means the wave will peak at 2 and dip to -2 from the center or midline of the graph.

The amplitude is always a positive value. It simply indicates the distance from the midpoint of the wave to either its peak or trough.

• Maximum height from the center: 2
• Maximum depth from the center: -2

Period

The period of a sine function is the distance along the x-axis before the function begins to repeat its pattern. For the equation \( y = -2 \sin(3x - \pi) \), the period is defined by the formula \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) within the sine function. In this case, \( B = 3 \), so the period becomes \( \frac{2\pi}{3} \).

This means that after every \( \frac{2\pi}{3} \) units along the x-axis, the sine wave will complete a full cycle and start anew. Periods help in predicting when the pattern of the wave will recur and are crucial for graph sketching and understanding wave behavior.

• Period Formula: \( \frac{2\pi}{B} \)
• Calculated Period: \( \frac{2\pi}{3} \)

Phase Shift

The phase shift of a function refers to a horizontal translation along the x-axis. It determines where the pattern of the sine wave begins. In the function \( y = -2 \sin(3x - \pi) \), the phase shift is calculated by dividing the constant \( C \) by the coefficient \( B \).Here, \( C = \pi \) and \( B = 3 \), thus the phase shift is \( \frac{\pi}{3} \) to the right.

A positive phase shift indicates a shift to the right, while a negative phase shift indicates a shift to the left. Phase shifts are vital as they show the starting position of the wave's cycle on the graph.

• Phase Shift Formula: \( \frac{C}{B} \)
• Calculated Phase Shift: \( \frac{\pi}{3} \) (right)

Sine Function

The sine function is a smooth oscillating function that forms a wave-like pattern. It is fundamental within trigonometry and appears frequently in various scientific disciplines including physics and engineering.

For the general sine function \( y = A \sin(Bx + C) \), it takes the form of an ongoing wavy pattern with key characteristics:

• Amplitude: Determines wave height (positive or negative).
• Period: Dictates length of one complete wave cycle.
• Phase Shift: Shows horizontal position of the sine pattern.

In our example, the sine function \( y = -2\sin(3x - \pi) \) forms a wave that flips below the x-axis, due to its negative amplitude, starting from its minimum point at a phase-shifted beginning.

Graph Sketching

Graph sketching is the process of portraying a function visually on a coordinate plane. For the sine function \( y = -2 \sin(3x - \pi) \), we follow distinct steps to create an accurate representation.

Start by identifying and marking the phase shift: \( \frac{\pi}{3} \) to the right. This means that the starting point of the sine wave cycle is shifted from the origin. Next, conceptualize the period \( \frac{2\pi}{3} \), the completion length of one cycle. Lastly, reflect the negative amplitude by beginning at -2 on the y-axis, representing the function's minimum point.

• Mark phase shift: \( \frac{\pi}{3} \) right on x-axis.
• Identify cycle length: \( \frac{2\pi}{3} \)
• Begin wave at minimum point due to negative amplitude.
• Sketch wave within amplitude range: -2 to 2.

Graphing visually brings the function to life and aids in better understanding wave characteristics and transformations.