Problem 38 - Precalculus Mathematics for Calculus

Question

Find the amplitude, period, and phase shift of the function, and graph one complete period. $$y=1+\cos \left(3 x+\frac{\pi}{2}\right)$$

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(\frac{2\pi}{3}\), Phase Shift: \(-\frac{\pi}{6}\).

Step 1: Identify the Standard Form

The standard form for a cosine function is \( y = a + b \cos(c(x - d)) \). Here, \( a = 1 \), \( b = 1 \), \( c = 3 \), and the phase shift is given in terms of addition as \( +\frac{\pi}{2} \). Thus, the function is \( y = 1 + \cos(3x + \frac{\pi}{2}) \).

Step 2: Determine the Amplitude

Amplitude of a cosine function is the absolute value of \( b \). In this function, \( b = 1 \), so the amplitude is \( |1| = 1 \).

Step 3: Compute the Period

The period of a cosine function is given by the formula \( \frac{2\pi}{|c|} \). Here, \( c = 3 \), so the period is \( \frac{2\pi}{3} \).

Step 4: Calculate Phase Shift

The phase shift can be found using \( \frac{-\text{phase constant}}{c} \). The phase constant here is \( \frac{\pi}{2} \), so the phase shift is \( \frac{-\frac{\pi}{2}}{3} = -\frac{\pi}{6} \). The graph shifts \( \frac{\pi}{6} \) units to the left.

Step 5: Graph One Complete Period

To graph the function, start by plotting the key points of one period, which spans from \( x = -\frac{\pi}{6} \) to \( x = -\frac{\pi}{6} + \frac{2\pi}{3} \). Mark the maximum at \( (1 + 1) = 2 \) and minimum at \( (1 - 1) = 0 \), drawing a cosine shape from 2 to 0, back to 2 over the given x-range.

Key Concepts

Amplitude

The amplitude of a trigonometric function refers to the height of the wave from its central axis. In simpler terms, it is how high and low the function goes from its average value. For the cosine function given in the exercise, the amplitude is determined by the coefficient in front of the cosine term. This coefficient is represented as \( b \) in the standard form \( y = a + b \cos(c(x - d)) \).
For the function \( y = 1 + \cos(3x + \frac{\pi}{2}) \), the coefficient \( b \) is 1. Hence, the amplitude is \( |b| = |1| = 1 \). This means that the graph of the function will rise and fall 1 unit above and below the central axis which is the horizontal line at \( y = a = 1 \).
Remember:

• The amplitude is always a positive value, as it is a distance measurement.
• It tells us the maximum variation of the function relative to its average value.

Period of Trigonometric Functions

The period of a trigonometric function is how long it takes for the function to repeat its shape. For sine and cosine functions, the period is typically \( 2\pi \) in their basic forms. However, the function can "speed up" or "slow down" depending on the coefficient \( c \) in the term \( c(x - d) \).
In the standard form \( y = a + b\cos(c(x - d)) \), the period is specifically calculated as \( \frac{2\pi}{|c|} \).
For our function \( y = 1 + \cos(3x + \frac{\pi}{2}) \), the value of \( c \) is 3. This gives us a period of:

• \( \frac{2\pi}{3} \)

This means the entire wave-like cycle (from peak, to trough, and back to peak) happens every \( \frac{2\pi}{3} \) units along the x-axis which is shorter than the normal cycle of \( 2\pi \). This compression of the cycle explains why the graph may appear more frequent or rapid as it oscillates.

Phase Shift

Phase shift refers to the horizontal movement of a trigonometric function along the x-axis. It is determined by the shift needed to align the starting point of one period with either the origin or any designated point as described by \( d \) in the function \( y = a + b\cos(c(x - d)) \).
As for the function \( y = 1 + \cos(3x + \frac{\pi}{2}) \), the phase term here is given as \( +\frac{\pi}{2} \). To find the phase shift, use the formula:

• \( \frac{-\text{phase constant}}{c} \)

Plugging in the values, the phase shift is \( \frac{-\frac{\pi}{2}}{3} = -\frac{\pi}{6} \). This means that the graph of the function is shifted to the left by \( \frac{\pi}{6} \) units.
This shift alters where the cycle starts compared to the basic cosine wave, essentially sliding the graph horizontally without affecting its amplitude or period.

Graphing Trigonometric Functions

Graphing a trigonometric function involves highlighting the movement and patterns that a function exhibits over its period. For \( y = 1 + \cos(3x + \frac{\pi}{2}) \), we begin by noting the key characteristics: amplitude, period, and phase shift.
The steps to graph the function are:

• Determine the amplitude (1 unit above and below the line \( y = 1 \)).
• Calculate the period (one complete cycle from \( x = -\frac{\pi}{6} \) to \( x = -\frac{\pi}{6} + \frac{2\pi}{3} \)).
• Adjust for a phase shift of \( -\frac{\pi}{6} \), which shifts the curve left.
• Plot key points within one period starting at the phase shift, marking maximum, minimum, and midline intersections.

Drawing these points ensures one complete wave is created by connecting them smoothly. Remember, the graph should show a regular cosine wave pattern adjusted by our calculated amplitude, period, and phase shift.