Problem 168 - 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. $$y=3 \cos (2 x+3)$$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \( \pi \), Phase shift: \( \frac{3}{2} \) units left.

Step 1: Identify the standard form

The standard form of a cosine function is \( y = A \cos(B(x - C)) + D \). Comparing this to our given function \( y = 3 \cos(2x + 3) \), we notice that \( A = 3 \), \( B = 2 \), and the expression is \( 2x + 3 \). This needs to be rewritten to find phase shift.

Step 2: Find the amplitude

The amplitude is the absolute value of \( A \). In our function, \( A = 3 \). Thus, the amplitude is \(|3| = 3\).

Step 3: Rewrite for phase shift

Rewrite \( 2x + 3 \) in terms of \( (x - C) \) by factoring out the 2: \( 2(x + \frac{3}{2}) \). Thus, \( C = -\frac{3}{2} \).

Step 4: Determine the period

The period of a cosine function is \( \frac{2\pi}{B} \). Here, \( B = 2 \). Thus, the period is \( \frac{2\pi}{2} = \pi \).

Step 5: Calculate the phase shift

The phase shift is determined by the value \( C \). Since \( C = -\frac{3}{2} \), the phase shift is \( \frac{3}{2} \) units to the left because the equation is \( x + \frac{3}{2} \).

Key Concepts

Understanding Amplitude

The amplitude of a trigonometric function, such as a cosine or sine wave, represents the height or the peak level from the midline of the graph. In simpler terms, it is the maximum distance the function rises or falls from its average value.

• For the standard function form, \(y = A \cos(B(x - C)) + D\), the amplitude is given by the absolute value of \(A\).
• It determines how "tall" or "short" the wave appears.

In our exercise with the function \(y = 3 \cos(2x + 3)\), the amplitude is \(3\). This means the maximum and minimum values are \(3\) units above and below zero, respectively. Knowing the amplitude allows you to predict the vertical stretch and compression of the wave.

Whenever you're analyzing the amplitude, remember it's always a positive number! This attribute does not change with horizontal shifts or changes to phase or frequency—only the vertical scaling considered in the function determines the amplitude.

Deciphering the Period

The period of a trigonometric function is the horizontal length required for the graph to complete one full cycle. Understanding the period helps in determining how frequently the waveform repeats itself across the x-axis.

• For functions like the cosine \(y = A \cos(B(x - C)) + D\), the period is calculated as \(\frac{2\pi}{B}\).
• It indicates how "stretched out" or "compressed" a wave appears horizontally.

In our particular exercise, since \(B = 2\), the period is \(\frac{2\pi}{2} = \pi\). This means the function reaches its full cycle in a span of \(\pi\) units along the x-axis.

Therefore, when observing the graph, every \(\pi\) units on the x-axis will show a repeated pattern. Checking the period helps with predicting how many complete cycles will appear in a given interval.

Getting a Grip on Phase Shift

The phase shift is essentially the horizontal movement of a trigonometric graph along the x-axis. This shift helps to determine where the function starts its cycle relatively to the standard position.

• In the standard cosine function \(y = A \cos(B(x - C)) + D\), \(C\) indicates the shift.
• If \(C\) is positive, the graph shifts to the right, if negative, it moves to the left.

In the exercise function \(y = 3 \cos(2x + 3)\), we rewrite \(2x + 3\) to the form \(2(x + \frac{3}{2})\). Here, \(C = -\frac{3}{2}\), translating to a phase shift of \(\frac{3}{2}\) units to the left.

Understanding phase shifts is crucial for predicting how a wave starts across the x-axis and visualizing how it has been moved horizontally from the standard position. It's a vital part of the function's horizontal alignment and symmetry.