Problem 56 - Precalculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify Amplitude

The amplitude of a cosine function in the form \( y = a \cos(bx + c) \) is the absolute value of \( a \). For the given function \( y = -4 \cos(2x - \pi) \), the value of \( a \) is -4. Thus, the amplitude is \( |-4| = 4 \).

Step 2: Determine the Period

The period of a cosine function is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). For the function \( y = -4 \cos(2x - \pi) \), \( b = 2 \). Hence, the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Calculate the Phase Shift

The phase shift of the cosine function is determined by solving \( bx + c = 0 \). The given function is \( y = -4 \cos(2x - \pi) \). Setting \( 2x - \pi = 0 \), we solve for \( x \): \( 2x = \pi \) thus \( x = \frac{\pi}{2} \). Therefore, the phase shift is \( \frac{\pi}{2} \) to the right.

Key Concepts

Amplitude

When discussing trigonometric functions like sine and cosine, amplitude is a key concept. It tells us how "tall" or "short" the wave of the function is. In simpler words, amplitude measures how far the graph of the function moves away from its central horizontal axis line.
For a cosine function in the form \( y = a \cos(bx + c) \), the amplitude is given by the absolute value of \( a \). It shows the maximum distance the function reaches from its equilibrium position, also known as the midline.

In our example function \( y = -4 \cos(2x - \pi) \):

• Here, the value of \( a \) is \(-4\).
• Remember, amplitude is always positive, so take the absolute value: \(||-4|| = 4\).

Therefore, this function has an amplitude of 4, which means the wave of this cosine function reaches 4 units above and below its central line. Amplitude is crucial because it tells us about the energy or "strength" of the wave.

Period

The period of a trigonometric function determines how long it takes for the function to complete one full cycle of its wave. Think of it like the time it takes for the wave to repeat its pattern.
For any cosine function \( y = a \cos(bx + c) \), the period is calculated using the formula \( \frac{2\pi}{b} \). This formula helps show that the value of \( b \) affects how stretched or compressed the wave looks along the x-axis.

In our specific function \( y = -4 \cos(2x - \pi) \):

• Identify \( b \), which is \( 2 \) here.
• Use the formula: \( \frac{2\pi}{2} = \pi \).

This means the period is \( \pi \). Understanding the period is essential as it tells you how many waves will fit in a given interval of the x-axis.For example, within a distance of \( \pi \) units along the x-axis, you'll see one complete wave of this function.

Phase Shift

Phase shift refers to the horizontal movement of the graph of a trigonometric function along the x-axis. This tells us where the wave starts relative to the y-axis.
To find the phase shift for a cosine function like \( y = a \cos(bx + c) \), we set \( bx + c = 0 \) and solve for \( x \). This reveals how much the function is shifted either to the left or to the right.

Using the provided function \( y = -4 \cos(2x - \pi) \):

• Set the inside of the cosine function, \( 2x - \pi \), equal to zero.
• Solve the equation: \( 2x = \pi \)
• So, \( x = \frac{\pi}{2} \).

Here, the phase shift is \( \frac{\pi}{2} \) to the right. This tells us that, compared to the standard cosine function, this graph starts its first peak further along the x-axis.Understanding phase shifts is important because they guide you in predicting where a function's pattern begins and ends.