Problem 44 - Precalculus with Calculus Previews

Question

Find the amplitude, period, and phase shift of the given function. Sketch at least one cycle of the graph. $$ y=4 \cos \left(2 x-\frac{3 \pi}{2}\right) $$

Step-by-Step Solution

Answer

Amplitude: 4, Period: \(\pi\), Phase Shift: \(\frac{3\pi}{4}\) to the right. Graph one cycle from \(\frac{3\pi}{4}\) to \(\frac{7\pi}{4}\).

Step 1: Identify the Amplitude

The given function is \[ y = 4 \cos\left(2x - \frac{3\pi}{2}\right) \] in the form \[ y = A \cos(Bx - C) \]where the amplitude \( A = 4 \).The amplitude is determined by the absolute value of \( A \), so the amplitude is 4.

Step 2: Calculate the Period

The formula for the period of a cosine function is given by \[ \text{Period} = \frac{2\pi}{|B|} \] where \( B \) is the coefficient of \( x \) in the argument of the cosine function. Here, \( B = 2 \).Substitute this value into the formula:\[ \text{Period} = \frac{2\pi}{2} = \pi \].Thus, the period is \( \pi \).

Step 3: Determine the Phase Shift

The phase shift is determined by the formula \[ \text{Phase Shift} = \frac{C}{B} \] where \( C \) is the constant inside the cosine function with \( Bx \).In the equation, \( C = \frac{3\pi}{2} \) and \( B = 2 \). Calculate the phase shift:\[ \text{Phase Shift} = \frac{\frac{3\pi}{2}}{2} = \frac{3\pi}{4} \].This phase shift indicates a horizontal shift \( \frac{3\pi}{4} \) to the right.

Step 4: Sketch One Cycle

To sketch one cycle, consider the characteristics of the function:1. **Amplitude:** 4, meaning the cosine curve oscillates between -4 and 4.2. **Period:** \( \pi \), meaning one full cycle occurs over an interval of \( \pi \) along the x-axis.3. **Phase Shift:** \( \frac{3\pi}{4} \) to the right, changing the starting point of the cycle.Label the x-axis from \( 0 \) to \( \pi \) while adjusting for the phase shift:- Start the cycle at \( x = \frac{3\pi}{4} \) and the end at \( x = \frac{3\pi}{4} + \pi = \frac{7\pi}{4} \).- The graph starts at maximum (4), then goes to zero, minimum (-4), back to zero, and finally returns to maximum within \( \pi \).

Step 5: Verify and Annotate

Verify that all labeled points conform to a cosine function’s standard behavior. Annotate the graph with critical points:- Maximum at \( \frac{3\pi}{4} \): Point (\( \frac{3\pi}{4}, 4 \))- Crosses x-axis at \( \frac{5\pi}{4} \) and \( \frac{7\pi}{4} \)- Minimum at \( \frac{3\pi}{2} \): Point (\( \frac{3\pi}{2}, -4 \))

Key Concepts

Phase Shift

The phase shift of a trigonometric function tells us how the function's graph is horizontally shifted relative to its standard position. It's an important concept to grasp because it affects the graph's starting point on the x-axis.
In our function, \[ y = 4 \cos\left(2x - \frac{3\pi}{2}\right) \] the phase shift is calculated using the formula:

• \( \text{Phase Shift} = \frac{C}{B} \)

Here, \( C = \frac{3\pi}{2} \) and \( B = 2 \). By substituting these values in, we get:

• \( \text{Phase Shift} = \frac{\frac{3\pi}{2}}{2} = \frac{3\pi}{4} \)

This positive phase shift suggests a rightward shift by \( \frac{3\pi}{4} \). This means that the entire cosine wave, which normally starts at zero, begins at \( x = \frac{3\pi}{4} \) along the x-axis.
Understanding phase shifts is crucial in modifying the starting points of wave patterns, which is especially useful in signal processing and harmonic motion analysis.

Cosine Function

The cosine function is one of the primary trigonometric functions and often appears in mathematical and scientific contexts. When we consider the standard form of a cosine function:\[ y = A \cos(Bx - C) \]several key parts influence its shape and position.

• **Amplitude**: Determined by \(|A|\). In our case, this is 4, indicating the function will oscillate between -4 and 4.
• **Period**: Calculated using \( \frac{2\pi}{|B|} \). For our function, \( B = 2 \), giving the period \( \pi \), meaning the pattern repeats every \( \pi \) units along the x-axis.
• **Phase Shift**: Already discussed, it alters the starting point of the function.

The cosine function inherently starts at its maximum value when there is no phase shift, and moves to zero, reaches a minimum, returns to zero, and ends the cycle where it started. This gives it its wave-like appearance.

Graphing Trigonometric Functions

Graphing trigonometric functions like the cosine function requires an understanding of amplitude, period, and phase shift. Let's look at how to plot one cycle of our given function:\[ y = 4 \cos\left(2x - \frac{3\pi}{2}\right) \]

• **Start Point**: Due to a phase shift of \( \frac{3\pi}{4} \) to the right, the cycle will start at \( x = \frac{3\pi}{4} \).
• **Interval**: Over a period of \( \pi \), the cycle extends to \( x = \frac{7\pi}{4} \).
• **Amplitude Consideration**: The maximum and minimum values reached are 4 and -4 respectively.

By marking critical points:

• The graph begins at the peak point \((\frac{3\pi}{4}, 4)\).
• It crosses the x-axis at \( \frac{5\pi}{4} \).
• The minimum point occurs at \( (\frac{3\pi}{2}, -4) \).
• Finally, it returns to zero at \( \frac{7\pi}{4} \)

When graphed, the curve shows a smooth wave, rising and falling between its amplitude boundaries, and shifted horizontally as determined by the phase shift. Understanding how these factors come together offers a complete picture of the function's behavior over one cycle.