Problem 31 - Precalculus Mathematics for Calculus

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$y=-2 \sin \left(x-\frac{\pi}{6}\right)$$

Step-by-Step Solution

Answer

Amplitude: 2, Period: \(2\pi\), Phase Shift: \(\frac{\pi}{6}\) right.

Step 1: Identify the Amplitude

The amplitude of a sine function in the form \(y = a \sin(b(x - c))\) is given by \(|a|\). In this case, the amplitude is \(|-2| = 2\).

Step 2: Determine the Period

The period of a sine function is calculated using the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) inside the sine function. Here, \(b = 1\), so the period is \(\frac{2\pi}{1} = 2\pi\).

Step 3: Calculate the Phase Shift

The phase shift can be found using \(c\) from the formula. Here, \(c = -\frac{\pi}{6}\), but because the equation is \(x - \frac{\pi}{6}\), the phase shift is \(\frac{\pi}{6}\) to the right.

Step 4: Graph One Complete Period

Use the values calculated: The amplitude is 2, meaning the graph oscillates between -2 and 2. The period is \(2\pi\), so label the x-axis from \(0\) to \(2\pi\). Begin plotting at \(x = \frac{\pi}{6}\) due to the phase shift. Mark the points at intervals of \(\frac{2\pi}{4} = \frac{\pi}{2}\) starting from \(\frac{\pi}{6}\) to complete one cycle. Ensure the wave peaks and troughs correspond to the amplitude.

Key Concepts

Amplitude

In a sine function, the amplitude refers to how far the graph stretches vertically from its center line. It is determined by the coefficient in front of the sine function itself. For the equation given, \( y = -2 \sin\left(x - \frac{\pi}{6}\right) \), the amplitude is represented by the absolute value \(|-2|\). Hence, the amplitude is 2.

This means the graph of this sine function will oscillate between +2 and -2 on the vertical axis. In practical terms, the amplitude tells us how tall the wave is. It defines the wave’s height from its midpoint to its peak or trough.

To visualize:

• The peak (highest point) is at +2.
• The trough (lowest point) is at -2.

Understanding the amplitude is crucial when sketching the graph because it shows the range within which the entire wave oscillates.

Period

The period of a sine function describes the horizontal length of one complete cycle of the wave. For the sine function \( y = -2 \sin\left(x - \frac{\pi}{6}\right) \), the period is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). Here, \( b = 1 \), so the period is calculated as \( \frac{2\pi}{1} = 2\pi \).

This tells us that it takes a distance of \( 2\pi \) on the x-axis for the pattern to repeat itself.

Visually on the graph:

• The wave makes a complete cycle from 0 to \( 2\pi \) on the x-axis.
• Within this span, all distinctive features such as crests and troughs occur before starting over.

Recognizing the period helps in properly spacing and scaling the graph to show the full oscillation.

Phase Shift

The phase shift in a sine function tells us how the wave is shifted from its standard position horizontally, either to the left or right. For the equation \( y = -2 \sin\left(x - \frac{\pi}{6}\right) \), the phase shift is determined by the value of \( c \) in the generalized function, defined as \( y = a \sin(b(x - c)) \). Here, \( c \) is \( \frac{\pi}{6} \), which signifies a shift of \( \frac{\pi}{6} \) units to the right.

This horizontal shift means:

• The starting point of the sine wave moves from where it typically is at \( x = 0 \) to \( x = \frac{\pi}{6} \).
• The entire graph is adjusted so that every feature occurs \( \frac{\pi}{6} \) units later on the x-axis than it would without the phase shift.

Understanding the phase shift is crucial for finding the correct starting point for the graph.

Sine Function

The sine function is one of the fundamental trigonometric functions often represented as \( y = a \sin(b(x - c)) + d \). This function describes how wave-like patterns repeat over intervals, making it essential in modelling various cyclic phenomena.

In a typical sine function without transformations, there are a few important things to note:

• It oscillates between +1 and -1.
• The basic period is \( 2\pi \).
• There are no phase shifts or vertical displacements.

However, in the function \( y = -2 \sin\left(x - \frac{\pi}{6}\right) \):

• The amplitude changes to 2, meaning our wave stretches to 2 and -2.
• There is a phase shift of \( \frac{\pi}{6} \), meaning the entire wave starts further along the x-axis.
• The period remains at \( 2\pi \) because the coefficient \( b \) is 1.

Grasping these changes helps us not just graph them accurately but also understand how different parameters of a sine function modify its behavior.