Problem 57 - Precalculus

Question

In Exercises \(49-60\), state the amplitude, period, and phase shift (including direction) of the given function. $$y=-\frac{1}{4} \cos \left(\frac{1}{4} x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

Amplitude: \(\frac{1}{4}\), Period: \(8\pi\), Phase Shift: \(2\pi\) right.

Step 1: Identify the Form of the Function

The function given is of the form \( y = a \cos(bx - c) \). In this case, \( a = -\frac{1}{4} \), \( b = \frac{1}{4} \), and \( c = \frac{\pi}{2} \).

Step 2: Determine the Amplitude

The amplitude of a cosine function \( y = a \cos bx \) is given by the absolute value of \( a \). Thus, the amplitude is \( \left| -\frac{1}{4} \right| = \frac{1}{4} \).

Step 3: Calculate the Period

The period of a cosine function is \( \frac{2\pi}{b} \). Here, \( b = \frac{1}{4} \), so the period is \( \frac{2\pi}{\frac{1}{4}} = 8\pi \).

Step 4: Find the Phase Shift

The phase shift of \( y = a \cos(bx - c) \) is given by \( \frac{c}{b} \). Substituting in the values gives \( \frac{\pi/2}{\frac{1}{4}} = 2\pi \). The function is shifted to the right by \( 2\pi \).

Key Concepts

Amplitude

The amplitude of a trigonometric function, like our cosine function, relates to its maximum height. It shows how far the peaks and valleys of the wave are from the center line (usually the horizontal axis). To find the amplitude of the function \(y = a \cos(bx - c)\), you need to take the absolute value of \(a\). In our example function \(y = -\frac{1}{4} \cos\left(\frac{1}{4}x - \frac{\pi}{2}\right)\), the amplitude is \(|-\frac{1}{4}|\), which simplifies to \(\frac{1}{4}\).

This shows that the function oscillates between \(\frac{1}{4}\) above and \(\frac{1}{4}\) below the center line. Amplitude is always positive, as it represents the distance, not the direction of the wave.

Period

The period of a trigonometric function indicates the length of one complete cycle of its wave. For the cosine function, the default period is \(2\pi\), but this changes with the introduction of a coefficient before \(x\). The formula to calculate the period is \(\frac{2\pi}{b}\), where \(b\) is the coefficient in front of \(x\) in the functions' equation.

In our function \(y = -\frac{1}{4} \cos\left(\frac{1}{4}x - \frac{\pi}{2}\right)\), \(b = \frac{1}{4}\). By plugging this into the formula, the period becomes \(\frac{2\pi}{\frac{1}{4}} = 8\pi\). This means the wave completes one full cycle every \(8\pi\) units on the x-axis, which is a stretch compared to the standard cosine wave.

Phase Shift

The phase shift tells us how much a function is shifted horizontally from its usual position. For the function \(y = a \cos(bx - c)\), the phase shift is calculated by \(\frac{c}{b}\). The phase shift also indicates the direction: if the result is positive, the shift is to the right; if negative, to the left.

Within our function \(y = -\frac{1}{4} \cos\left(\frac{1}{4}x - \frac{\pi}{2}\right)\), \(c = \frac{\pi}{2}\) and \(b = \frac{1}{4}\). The calculation \(\frac{\pi/2}{\frac{1}{4}} = 2\pi\) means the phase shift is \(2\pi\) units to the right. This horizontal shift adjusts where each cycle of the wave starts.