Problem 167 - Calculus Volume 1

Question

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: \(\frac{\pi}{4}\) to the right.

Step 1: Identify the amplitude

The amplitude of a sine function in the form of \(y = a \sin(bx - c) + d\) is given by the absolute value of \(a\). In this equation \(y = \sin\left(x - \frac{\pi}{4}\right)\), \(a = 1\). Therefore, the amplitude is 1.

Step 2: Find the period

The period of a sine function \(y = a \sin(bx - c) + d\) is given by \(\frac{2\pi}{b}\). In this equation, since the coefficient of \(x\) (i.e., \(b\)) is 1, the period is \(\frac{2\pi}{1} = 2\pi\).

Step 3: Determine the phase shift

The phase shift is determined by the expression \(\frac{c}{b}\) where \(y = a \sin(bx - c) + d\). Here, \(c = \frac{\pi}{4}\) and \(b = 1\), giving a phase shift of \(\frac{\pi}{4}\). Since this is subtracted from \(x\), the phase shift is \(\frac{\pi}{4}\) to the right.

Key Concepts

Amplitude

The amplitude of a sine function tells us how tall or short each wave in the function will be. It represents the height from the midline of the wave to its peak (or trough). In the general form of a sine function, which is \(y = a \sin(bx - c) + d\), the amplitude is the absolute value of \(a\). This means we ignore any negative sign in front of \(a\) because amplitude is always positive.

For the exercise function \(y = \sin\left(x - \frac{\pi}{4}\right)\), the value of \(a\) is 1 because no coefficient is written in front of the sine function, which implies an invisible 1. Therefore:

• Amplitude = |1| = 1

So, the waves of this sine function will rise and fall to a height of 1 unit above and below the midline, which is typically the x-axis unless shifted by another constant \(d\) in the equation.

Period

The period of a sine function refers to the length of one complete cycle of the wave before it repeats itself. This is important for understanding the frequency of the wave over a given range. To calculate the period of a sine function, we use the formula \(\frac{2\pi}{b}\), extracted from the general form \(y = a \sin(bx - c) + d\), where \(b\) influences how stretched or compressed the wave will be.

For the function \(y = \sin\left(x - \frac{\pi}{4}\right)\):

• \(b = 1\), which compels us to use the formula \(\frac{2\pi}{1} = 2\pi\)

This means that the wave starts repeating every \(2\pi\) units along the x-axis. Since \(b = 1\), the wave takes the standard cycle length, similar to the basic sine curve. A higher \(b\) would result in more cycles within the same range, while a smaller \(b\) (between 0 and 1) would stretch the cycles out.

Phase Shift

The phase shift tells us how much the standard sine wave is shifted horizontally along the x-axis. It indicates whether the wave starts sooner or later in relation to the y-axis.
To find this shift in the general sine function form \(y = a \sin(bx - c) + d\), use the formula \(\frac{c}{b}\). The direction of the shift depends on the sign of \(c\): subtracting \(c\) shifts the function to the right, whereas adding shifts it to the left.
For \(y = \sin\left(x - \frac{\pi}{4}\right)\):

• Here, \(c = \frac{\pi}{4}\), and \(b = 1\)
• Phase Shift = \(\frac{\frac{\pi}{4}}{1} = \frac{\pi}{4}\)

This results in a shift of \(\frac{\pi}{4}\) units to the right. It's like sliding the whole wave towards the positive x-direction, which changes the starting point of the wave's ascent on the x-axis.