Problem 167 - Calculus Volume 1

Question

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function. $$y=\sin \left(x-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

Amplitude: 1; Period: \(2\pi\); Phase shift: right \(\frac{\pi}{4}\).

Step 1: Identify the amplitude

The amplitude of a sine function written in the form \( y = a \sin(bx-c) \) is the absolute value of \( a \). In our function \( y = \sin \left(x-\frac{\pi}{4}\right) \), the amplitude is the coefficient in front of the sine function, which is 1. Thus, the amplitude is 1.

Step 2: Determine the period

The period of a sine function \( y = a \sin(bx-c) \) is given by \( \frac{2\pi}{b} \). In this function, \( b = 1 \). Therefore, the period is \( \frac{2\pi}{1} = 2\pi \).

Step 3: Calculate the phase shift

The phase shift \( \phi \) of a sine function \( y = a \sin(bx - c) \) is determined by the expression \( \frac{c}{b} \). For this function, \( c = \frac{\pi}{4} \) and \( b = 1 \), so the phase shift is \( \frac{\pi}{4} \). Since the expression is \( x - \frac{\pi}{4} \), the phase shift is to the right by \( \frac{\pi}{4} \).

Key Concepts

Amplitude

When you hear the term "amplitude" in trigonometry, think of it as the height of the wave. Specifically, it's the maximum distance that the function can reach from its central axis (usually the x-axis in a graph of a function). In simple terms, amplitude represents the strength or intensity of the wave.
In trigonometric functions like sine and cosine, the amplitude is denoted by the coefficient in front of the sine function, written as \( y = a \sin(bx - c) \). For our function, \( y = \sin\left(x-\frac{\pi}{4}\right) \), we see no coefficient other than 1 because it is written simply as \( \sin \). Hence, the amplitude is \( |1| = 1 \).
To sum it up:

• The amplitude does not affect the frequency or the phase of the wave, only its height.
• In this function, because the amplitude equals 1, the wave reaches +1 and -1 from the horizontal axis.

Period

The period of a function is crucial because it tells us how long it takes for the wave cycle to repeat itself. Imagine drawing a single wave segment repeatedly on graph paper—the horizontal length of one complete wave is the period.
In mathematical terms, the period of a sine or cosine function \( y = a \sin(bx - c) \) is calculated by \( \frac{2\pi}{b} \). This formula arises from the nature of sine and cosine functions, which naturally complete a single cycle in \( 2\pi \) radians. For the given function, with \( b = 1 \), the period is \( \frac{2\pi}{1} = 2\pi \).
Here's what this means:

• The wave's pattern repeats every \( 2\pi \) units along the x-axis.
• This repetition is uniform and does not depend on the wave's height.

Understanding the period provides insight into the wave's frequency, or how often it repeats over a certain interval.

Phase Shift

A phase shift occurs when the wave starts its cycle at a point other than the usual zero point. You can think of it as the wave being shifted horizontally along the x-axis.
In the function \( y = a \sin(bx - c) \), the phase shift is calculated using \( \frac{c}{b} \). This determines how far and in which direction the wave has shifted. For our function, \( c = \frac{\pi}{4} \) and \( b = 1 \), giving a phase shift of \( \frac{\pi}{4} \).
By looking at the expression inside the sine function, \( x - \frac{\pi}{4} \), we can see the wave has been shifted to the right by \( \frac{\pi}{4} \) units.
This means:

• Without the phase shift, the sine wave would start at zero when \( x = 0 \).
• With this shift, the wave starts its upward swing a little later, at \( x = \frac{\pi}{4} \).

Recognizing phase shifts helps in predicting graph positions and understanding how the function behaves compared to its standard form.