Problem 7 - 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. $$ y=3 \sin \left(x+\frac{\pi}{6}\right) $$

Step-by-Step Solution

Answer

Amplitude = 3, Period = \(2\pi\), Phase Shift = \(-\frac{\pi}{6}\).

Step 1: Identify the Equation Form

The given equation is \( y = 3 \sin(x + \frac{\pi}{6}) \). This is of the form \( y = A \sin(B(x - C)) \), where \( A \) is the amplitude, \( B \) affects the period, and \( C \) is the phase shift.

Step 2: Determine the Amplitude

The amplitude of the function is the absolute value of \( A \). For the equation \( y = 3 \sin(x + \frac{\pi}{6}) \), \( A = 3 \). Thus, the amplitude is 3.

Step 3: Calculate the Period

The period of a sine function is given by \( \frac{2\pi}{B} \). In our equation, \( B = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \). Thus, the period is \( 2\pi \).

Step 4: Find the Phase Shift

The phase shift is determined by removing the term inside the parentheses \( (x + \frac{\pi}{6}) \) and identifying \( C \). Here, \( C = -\frac{\pi}{6} \) because we rewrite it as \( x - (-\frac{\pi}{6}) \). So, the phase shift is \( -\frac{\pi}{6} \), which means a shift to the left by \( \frac{\pi}{6} \).

Step 5: Sketch the Graph

Based on the amplitude, period, and phase shift, sketch the sine function. Start at \( -\frac{\pi}{6} \) on the x-axis due to the phase shift, showing peaks at 3, troughs at -3, and completing one full cycle in \( 2\pi \) units. Ensure the midline is unchanged at y = 0.

Key Concepts

Amplitude

When talking about the amplitude of a sine function, we are referring to its height. Essentially, it's how high or low the wave reaches from its central axis.
In mathematical terms, the amplitude is the absolute value of the coefficient in front of the sine function.

• For the given equation, \( y = 3 \sin(x + \frac{\pi}{6}) \), the coefficient is 3. Thus, the amplitude is \(|3| = 3\).
• Amplitude is always a non-negative value.
• It tells us that the wave will reach a maximum of 3 units above and a minimum of 3 units below the horizontal axis (y = 0), which is the midline.

This means the highest point of the wave (peak) is at 3, and the lowest point (trough) is at -3. Understanding amplitude helps visualize how "tall" or "deep" a wave is, which is key when graphing sine functions.

Period

The period of a sine function is the length of one complete cycle of the wave. It signifies how long it takes for the sine wave to start repeating itself.
The formula for finding the period is \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the sine function.

• In our specific example, \( y = 3 \sin(x + \frac{\pi}{6}) \), the coefficient \(B\) is 1.
• Therefore, the period is \(\frac{2\pi}{1} = 2\pi\).

What this implies is that the sine function completes a full wave cycle from where it starts back to the same point in a distance of \(2\pi\) along the x-axis. When graphing, understanding the period helps place the wave correctly across the x-axis, ensuring accurate spacing for one complete oscillation.

Phase Shift

Phase shift refers to the horizontal movement of the sine wave along the x-axis. It is determined by the value inside the parentheses with the \(x\).
To calculate the phase shift, we look at the equation form \(y = A \sin(B(x - C))\) and solve for \(C\).

• In the function \( y = 3 \sin(x + \frac{\pi}{6}) \), we have \(x + \frac{\pi}{6}\) which we rewrite as \(x - (-\frac{\pi}{6})\).
• Thus, "C" is \(-\frac{\pi}{6}\), indicating the phase shift is \(-\frac{\pi}{6}\).

The negative value implies the sine wave is shifted to the left by \(\frac{\pi}{6}\) units on the x-axis.
Understanding phase shift is crucial for positioning the wave correctly, ensuring it starts and progresses from its exact new position horizontally.