Question

Find the amplitude (if one exists), period, and phase shift of each function. Graph each function. Be sure to label key points. Show at least two periods. $$ y=3 \cos (2 x+\pi) $$

Short answer

Amplitude: 3, Period: \( \pi \), Phase Shift: \(-\frac{\pi}{2}\)

Step-by-step solution

- Identify the amplitude

The amplitude of the function can be found by identifying the coefficient in front of the cosine function. In the equation \(y = 3 \cos(2x + \pi)\), the amplitude is the absolute value of 3. Therefore, the amplitude is 3.

- Determine the period

The period of the function can be determined using the formula \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the cosine function. Here, \(B = 2\). Hence, the period \(T\) is given by \[T = \frac{2\pi}{2} = \pi\].

- Calculate the phase shift

The phase shift can be determined using the formula \(\frac{-C}{B}\), where \(C\) is the constant inside the cosine function. Here, \(C = \pi\). Then the phase shift is \[ \frac{-(\pi)}{2} = -\frac{\pi}{2} \]. The phase shift is \( -\frac{\pi}{2} \), which means the graph is shifted \(\frac{\pi}{2} \) units to the left.

- Graph the function

To graph \(y = 3 \cos(2x + \pi)\), follow these steps:1. Start by plotting the cosine graph with the given amplitude of 3.2. The period is \( \pi\), so label your x-axis accordingly to show two complete periods from 0 to \(2\pi\).3. Apply the phase shift by moving the graph \( \frac{\pi}{2} \) units to the left.4. Identify and label key points such as the maximum, minimum, and intercepts within each period.

Key concepts

Amplitude

The **amplitude** of a trigonometric function, specifically in a cosine or sine function, affects its height. It measures how far the function's peaks and valleys are from the central axis, typically the x-axis. The pattern for amplitude is given by the coefficient in front of the cosine or sine terms.

For the function provided, \(y = 3 \, \cos{(2x + \pi)}\), the amplitude is directly derived from the given coefficient of 3. Thus, the peaks will reach up to 3 units above and below the x-axis.

The general rule is:

• If \(A\) is the multiplier in \(y = A \, \cos{(Bx + C)}\), then the amplitude is \(|A|\).
• For example, if \(A = 3\), the absolute value \(|3| = 3\) is the amplitude.

This means the cosine wave will vary between -3 and 3.

Period

The **period** of a function defines the length of one complete cycle of its graph. In trigonometric functions like cosine and sine, it shows how the function repeats itself periodically.

To determine the period of a cosine function given by \(y = 3 \, \cos{(2x + \pi)}\), use the formula \(T = \frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\) inside the cosine expression.

• Here, \(B = 2\).
• Substituting into the formula, the period \(T = \frac{2\pi}{2} = \pi\).

This indicates the function repeats itself every \(\pi\) units along the x-axis.

The period provides a framework to help plot the function correctly, showing the start and end of one cycle, and helps in labeling the graph accurately.

Phase Shift

The **phase shift** of a trigonometric function shifts the entire graph horizontally. It is determined by the value \(C\) in the function \(y = A \, \cos{(Bx + C)}\), using the formula \(\frac{-C}{B}\).

In the given function \(y = 3 \, \cos{(2x + \pi)}\), \(C = \pi\) and \(B = 2\).
Using the phase shift formula:

• \(\text{Phase shift} = \frac{-(\pi)}{2} = -\frac{\pi}{2}\)

The phase shift of \(-\frac{\pi}{2}\) means that the graph shifts \(\frac{\pi}{2}\) units to the left. This leftward shift relocates the standard starting points of maximum, minimum, and intercepts of the cosine function.

Understanding phase shifts lets you accurately move the graph horizontally without altering its shape or other properties such as amplitude and period.

Graphing Trigonometric Functions

Graphing trigonometric functions involves correctly interpreting their amplitude, period, and phase shift to plot an accurate graph.

To graph \(y = 3 \, \cos{(2x + \pi)}\):
1. Find the amplitude to determine the peak and valley (maximum and minimum values). Here, amplitude = 3.
2. Determine the period using \(\frac{2\pi}{B}\). Period = \(\pi\).
3. Calculate the phase shift with \(\frac{-C}{B}\). Phase shift = \(-\frac{\pi}{2}\).

Here's an easy method to graph:

• Start by plotting the basic shape of cosine function considering the amplitude. It fluctuates between 3 and -3.
• Set the x-axis over which one period (\(\pi\)) is extended. Show at least two periods by extending from 0 to \(2\pi\).
• Apply the phase shift and move the entire graph \(\frac{\pi}{2}\) units to the left.
• Label key points, such as maximum (3), minimum (-3), and intersections with the x-axis within each period.

This approach gives a clear and accurate depiction of how amplitude, period, and phase shift influence the trigonometric function graph.