Problem 171 - Calculus Volume 1

Question

Find a. the amplitude, b. the period, and c. the phase shift with direction for each function. \(y=-3 \sin (\pi x+2)\)

Step-by-Step Solution

Answer

Amplitude: 3; Period: 2; Phase shift: \(-\frac{2}{\pi}\) (left).

Step 1: Understanding the Amplitude

The amplitude of a sine function in the form of \( y = a \sin(bx + c) \) is given by \(|a|\). In this case, the given function is \(y = -3 \sin(\pi x + 2)\). Hence, \(a = -3\). The amplitude is therefore \(|-3| = 3\).

Step 2: Calculating the Period

The period of a sine function \(y = a \sin(bx + c)\) is given by \(\frac{2\pi}{|b|}\). In our function, \(b = \pi\). Therefore, the period is \(\frac{2\pi}{|\pi|} = 2\).

Step 3: Determining the Phase Shift

The phase shift of a sine function \(y = a \sin(bx + c)\) is calculated using \(-\frac{c}{b}\). Here, \(c = 2\) and \(b = \pi\). Thus, the phase shift is \(-\frac{2}{\pi}\).

Step 4: Direction of the Phase Shift

The phase shift \(-\frac{2}{\pi}\) is negative, indicating a shift to the left. Hence, the function is shifted \(\frac{2}{\pi}\) units to the left.

Key Concepts

Amplitude

Amplitude is a key feature in trigonometric functions, especially the sine function. It reflects the strength or intensity of the wave. Amplitude can be understood as how far the wave extends from its central line or axis in both directions. In any sine wave of the form \( y = a \sin(bx + c) \), the amplitude is determined by \(|a|\). It simply ignores the sign and uses the absolute value. This is because amplitude, being a measure of distance, is always positive.
For instance, given \( y = -3 \sin(\pi x + 2) \), we find \( a = -3 \). Thus, the amplitude is \( |-3| = 3 \). This tells us that our wave stretches 3 units above and below its central line.
So, amplitude helps us understand the height of the waves, which is crucial in analyzing the overall behavior of trigonometric functions.

Period

The period of a function is a core aspect that deals with how often the wave repeats itself over a length. In a sine function, the period can be calculated using the formula \( \frac{2\pi}{|b|} \), where \( b \) is the coefficient of \( x \) within the sine function \( y = a \sin(bx + c) \).
In the case of \( y = -3 \sin(\pi x + 2) \), we have \( b = \pi \). Substituting, the period becomes \( \frac{2\pi}{|\pi|} = 2 \).
This result means that every 2 units along the x-axis, the sine wave completes a full cycle. Easily predicting how often the graph of the function "restarts" its pattern is a huge advantage when working with trigonometric functions.

Phase Shift

Phase shift refers to the horizontal movement of the sine wave along the x-axis. It tells us where the wave starts. You can calculate it with the formula \( -\frac{c}{b} \) from the function \( y = a \sin(bx + c) \).
For the function \( y = -3 \sin(\pi x + 2) \), \( c = 2 \) and \( b = \pi \). Therefore, the phase shift is calculated as \( -\frac{2}{\pi} \).
This negative value indicates that the sine function is shifted \( \frac{2}{\pi} \) units to the left. Understanding phase shifts is crucial because it allows us to adjust and predict the behavior of the wave at different points on the x-axis.

Sine Function

The sine function is one of the most widely used trigonometric functions. It models periodic phenomena, like sound waves or tides, due to its wave-like nature. The standard form is \( y = a \sin(bx + c) \), where:

• \( a \) determines the amplitude, which reflects the wave's height.
• \( b \) affects the period, dictating how often the wave repeats.
• \( c \) shifts the wave, changing its starting point horizontally.

The function is known for its smooth, oscillating curve which moves above and below the x-axis. Each feature of the sine function allows for adjustment and manipulation to fit various real-world scenarios, making it crucial in both theoretical and applied mathematics.