Problem 55 - Precalculus

Question

In Exercises \(49-60\), state the amplitude, period, and phase shift (including direction) of the given function. $$y=3 \sin (2 x+\pi)$$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \( \pi \), Phase Shift: \( \frac{\pi}{2} \) left.

Step 1: Identify Amplitude

The amplitude of a sinusoidal function of the form \( y = A \sin(Bx + C) + D \) is given by the absolute value of \( A \). For the function \( y = 3 \sin (2x + \pi) \), \( A = 3 \), so the amplitude is 3.

Step 2: Calculate Period

The period of a sine function is calculated using the formula \( \text{Period} = \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) in the expression. Here, \( B = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Determine Phase Shift

The phase shift of a sinusoidal function is found using the formula \( \text{Phase Shift} = -\frac{C}{B} \), where \( C \) is the constant added inside the function. In our function, \( C = \pi \), and \( B = 2 \). Thus, the phase shift is \(-\frac{\pi}{2}\).

Step 4: Identify Direction of Phase Shift

Since the phase shift is negative, this means the graph shifts to the left by \( \frac{\pi}{2} \).

Key Concepts

Amplitude

Amplitude is the measure of how much a sine or cosine wave oscillates above and below its axis, often regarded as the "height" of the wave. In a trigonometric function of the form \(y = A \sin(Bx + C) + D\), the amplitude is represented by \(A\).

• In the function \(y = 3 \sin(2x + \pi)\), we identify \(A\) as 3.
• The absolute value of \(A\) gives us the amplitude: |3| = 3.

Therefore, the wave will peak at 3 units above and trough at 3 units below its midpoint. Understanding amplitude is crucial because it tells us how large the oscillations are, making it an essential characteristic of trigonometric functions often found in physical systems like sound waves and light waves.

Period

The period of a trigonometric function is the length required for the function to complete one full cycle. To find the period of a sine function given in the form \(y = A \sin(Bx + C) + D\), we use the formula \(\text{Period} = \frac{2\pi}{B}\).

• For our exercise function, \(y = 3 \sin(2x + \pi)\), \(B\) is equal to 2.
• Applying the formula, the period is computed as \(\frac{2\pi}{2} = \pi\).

This means that the sine wave pattern will repeat every \(\pi\) units along the x-axis. Recognizing the period is vital as it helps in understanding the frequency of oscillations, which is the reciprocal of the period. This is particularly useful in many fields such as signal processing, acoustics, and other wave-related studies.

Phase Shift

The phase shift of a trigonometric function indicates how much the graph of the function is shifted left or right from its standard position. We determine the phase shift for a function \(y = A \sin(Bx + C) + D\) using the formula \(\text{Phase Shift} = -\frac{C}{B}\).

• For \(y = 3 \sin(2x + \pi)\), \(B\) is 2 and \(C\) is \(\pi\).
• Plug these values into the formula to get \(-\frac{\pi}{2}\).

The negative sign indicates a shift to the left. Specifically, the graph moves \(\frac{\pi}{2}\) units left. Understanding phase shift is important for graphing functions as it helps to pinpoint where the cycle of the wave starts. This concept finds applications in fields like electronics and communications, where wave synchronization is critical.