Problem 59 - Precalculus

Question

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

Step-by-Step Solution

Answer

Amplitude: 2, Period: 4, Phase shift: 4 units right.

Step 1: Identify the Function Type

The given function is a cosine function of the form \( y = a \cos(b(x - c)) \). This helps in determining its amplitude, period, and phase shift.

Step 2: Determine the Amplitude

The amplitude of a trigonometric function \( y = a \cos(bx) \) is given by the absolute value of \( a \), which is the coefficient in front of the cosine. Here, \( a = 2 \). Thus, the amplitude is \( 2 \).

Step 3: Calculate the Period

For a cosine function \( y = a \cos(bx) \), the period is calculated using the formula \( \frac{2\pi}{b} \). Here, \( b = \frac{\pi}{2} \). Therefore, the period is \( \frac{2\pi}{\frac{\pi}{2}} = 4 \).

Step 4: Determine the Phase Shift

The phase shift of a cosine function \( y = a \cos(b(x-c)) \) is calculated by solving \( x = c \). Here, \( c = 4 \), which means the function is shifted 4 units to the right.

Key Concepts

Amplitude

Amplitude is a fundamental concept when working with trigonometric functions like sine or cosine. In simple terms, the amplitude measures how far the function's peaks and troughs (maximum and minimum points) extend from its central axis.

• For a cosine or sine wave, the amplitude is the absolute value of the coefficient found immediately in front of the trigonometric function.
• In the function \(y = 2 \cos\left[\frac{\pi}{2}(x-4)\right]\), the amplitude is 2. This tells us that the highest point (peak) of the wave is 2 units above the central axis, and the lowest point (trough) is 2 units below.

Each point of the wave oscillates 2 units from the center, giving us information about the function's vertical stretch. Amplitude does not have any direction associated with it, as it is always a positive number. It's essential to observe the absolute value of the coefficient to find the correct amplitude.

Period

The period of a trigonometric function defines how long it takes for the function to complete one full cycle. In other words, it is the horizontal length of one complete wave, from peak to peak or trough to trough.

• For a basic cosine function \(y = a \cos(bx)\), the period can be calculated using the formula \(\frac{2\pi}{b}\), where \(b\) is the coefficient multiplied by the variable \(x\) inside the cosine bracket.
• In the given function \(y = 2 \cos\left[\frac{\pi}{2}(x-4)\right]\), we identify \(b\) as \(\frac{\pi}{2}\).
• Thus, the period is \(\frac{2\pi}{\frac{\pi}{2}}\), which simplifies to 4.

This means that the wave repeats every 4 units along the horizontal axis. A shorter period would indicate more rapid oscillations, whereas a longer period would mean the waves are more spread out. Understanding the period is crucial for predicting the behavior of the function over its domain.

Phase Shift

Phase shift refers to the horizontal displacement of the wave from its standard position. It tells you whether the function is shifted left or right along the x-axis, starting from a known point.

• For functions of the form \(y = a \cos(b(x-c))\), the phase shift is determined by \((x-c)\). The term \(c\) dictates the direction and magnitude of the shift.
• In the function \(y = 2 \cos\left[\frac{\pi}{2}(x-4)\right]\), this value \(c\) is 4.
• A positive \(c\) value indicates the function shifts to the right, meaning the wave begins 4 units to the right on the x-axis compared to its standard position.

By understanding phase shifts, you gain insight into where a wave starts on the horizontal plane. This shift adjusts the starting point of the cycle without altering other characteristics like amplitude or period. It is an important factor when graphing trigonometric functions, ensuring they align with specific points as required.