Problem 172 - 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. $$y=4 \cos \left(2 x-\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

a. Amplitude = 4 b. Period = \(\pi\) c. Phase shift = \(\frac{\pi}{4}\) to the right

Step 1: Identify Amplitude

The amplitude of a cosine function, given in the form \(y = a \cos(bx - c)\), is the absolute value of \(a\). Here, \(a = 4\). Therefore, the amplitude is \(|4| = 4\).

Step 2: Determine the Period

To find the period of a cosine function given in the form \(y = a \cos(bx - c)\), we use the formula \(\frac{2\pi}{b}\). In the given function, \(b = 2\). Thus, the period is \(\frac{2\pi}{2} = \pi\).

Step 3: Calculate the Phase Shift

The phase shift for a cosine function \(y = a \cos(bx - c)\) is given by \(\frac{c}{b}\). Here, \(c = \frac{\pi}{2}\) and \(b = 2\). Therefore, the phase shift is \(\frac{\pi/2}{2} = \frac{\pi}{4}\). Since the expression is in the form \((bx - c)\), the shift is to the right.

Key Concepts

Amplitude of a Trigonometric Function

The amplitude of a trigonometric function measures its maximum height from the centerline. In a function like the cosine function, represented as \( y = a \cos(bx - c) \), the amplitude is the absolute value of the coefficient \( a \).
For example, if you have \( y = 4 \cos(2x - \frac{\pi}{2}) \), you can easily find the amplitude by noting the value of \( a \). Here, \( a = 4 \), so the amplitude is \( |4| = 4 \).
The amplitude tells us how "tall" the waves appear from the middle line, affecting how pronounced the peaks and valleys of the function are:

• An amplitude of \( 4 \) means the graph stretches or compresses vertically, reaching up to 4 units above and below the central axis.
• Amplitude does not affect the length of the cycle, but how much the cycle goes up and down.
• It's important to remember that amplitude is always a positive value since it represents a distance.

Period of a Function

The period of a trigonometric function is the horizontal length needed before the function starts repeating itself. It represents one complete cycle of the function. For cosine functions of the form \( y = a \cos(bx - c) \), the period is determined using the formula \( \frac{2\pi}{b} \).
In the case of our function \( y = 4 \cos(2x - \frac{\pi}{2}) \), the coefficient \( b = 2 \). Applying the formula, we get:

• \( \frac{2\pi}{2} = \pi \)

This tells us the function completes one full cycle every \( \pi \) units, instead of the usual \( 2\pi \) for a basic cosine function with \( b = 1 \).
The period affects how rapidly the waves of the function repeat:

• A smaller period leads to more frequent cycles within the same length.
• The change in period squeezes or stretches the graph horizontally.
• A larger \( b \) results in a smaller \( \frac{2\pi}{b} \), meaning more cycles fit in the same horizontal space.

Phase Shift of a Trigonometric Function

The phase shift of a trigonometric function is the horizontal shift left or right, relative to its regular position. You can think of it as moving the entire graph along the \( x \)-axis. The typical form for a cosine function is \( y = a \cos(bx - c) \), where the phase shift is calculated by \( \frac{c}{b} \).
For \( y = 4 \cos(2x - \frac{\pi}{2}) \), we observe that \( c = \frac{\pi}{2} \) and \( b = 2 \), meaning our calculation becomes:

• \( \frac{\frac{\pi}{2}}{2} = \frac{\pi}{4} \)

This results in a phase shift of \( \frac{\pi}{4} \), and since the equation takes the form \((bx - c)\), it indicates a shift to the right.
Understanding phase shifts can help you:

• Determine how far the wave's peaks and troughs have moved from the origin.
• Recognize the impact of phase shift alongside changes in period or amplitude.
• Predict the timing of cycle starts or significant changes in the function.

Whenever you graph a trigonometric function, these shifts are crucial for identifying exact placements on the \( x \)-axis.