Problem 36 - Algebra and Trigonometry

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 Amplitude

The given function is of the form \( y = a + \cos(bx+c) \). Here, the amplitude is determined by the coefficient of the cosine term, which is 1. Thus, the amplitude is \( |1| = 1 \).

Step 2: Determine the Period

The period of a cosine function \( \cos(bx) \) is \( \frac{2\pi}{|b|} \). For our function, \( b = 3 \). Therefore, the period is \( \frac{2\pi}{3} \).

Step 3: Calculate the Phase Shift

The phase shift is found using the term \( bx+c \). The phase shift is given by \( -\frac{c}{b} \). Here, \( c = \frac{\pi}{2} \) and \( b = 3 \), so the phase shift is \( -\frac{\frac{\pi}{2}}{3} = -\frac{\pi}{6} \).

Step 4: Graph One Complete Period

To graph the function, start at the phase shift \( x = -\frac{\pi}{6} \). A complete period ends at \( x = -\frac{\pi}{6} + \frac{2\pi}{3} \). Within this interval, plot key points (maxima, minima, intercepts) of the cosine function, considering the vertical shift up by 1 due to the \( +1 \) in the function.

Key Concepts

Amplitude

The amplitude of a trigonometric function, specifically in the form of a cosine or sine function, refers to the height of the wave's peaks from its midline. It essentially measures how much the function deviates from its average value or horizontal axis.
In the modified function\(y = 1 + \cos(3x + \frac{\pi}{2})\), the amplitude is determined by the coefficient of the cosine term. This coefficient is the value that precedes the cosine function mathematically. Here, the coefficient is 1.

• The amplitude tells us that the waveform for the standard cosine function is not stretched or compressed vertically.
• It's calculated as the absolute value: \( |1| = 1 \), so the amplitude is simply \(1\).

This amplitude signifies that from the midline, the function peaks reach a height of 1 unit.

Period

In trigonometry, the period of a function defines how long it takes for the function to repeat its pattern. To determine the period, one examines the coefficient of \(x\) in the cosine or sine function.
Our function is expressed as \(cos(3x + \frac{\pi}{2})\). The general period formula for any cosine function \(cos(bx)\) is given by \(\frac{2\pi}{|b|}\). Here, the coefficient \(b\) is 3.

• To find the period for this particular function, substitute \(b = 3\) into the period formula.
• The period becomes \(\frac{2\pi}{3}\).

This result means that every \(\frac{2\pi}{3}\) radians, the wave pattern of this trigonometric function will start over.

Phase Shift

Phase shift in a trigonometric function determines the horizontal displacement of the function's graph. Essentially, it shifts the graph horizontally to the left or right.
For the function \(y = 1 + \cos(3x + \frac{\pi}{2})\), the phase shift formula is \(-\frac{c}{b}\), where \(bx + c\) represents the inside part of the cosine function.
The values in our equation are \(c = \frac{\pi}{2}\) and \(b = 3\).

• Insert these into the phase shift formula: \(-\frac{\frac{\pi}{2}}{3} = -\frac{\pi}{6}\).
• This negative value signals a shift to the left by \(\frac{\pi}{6}\) units.

Hence, the entire graph of the function is moved \(\frac{\pi}{6}\) units to the left along the x-axis.

Graphing Trigonometric Functions

Graphing a trigonometric function such as \(y = 1 + \cos(3x + \frac{\pi}{2})\) involves several steps to ensure accuracy of its amplitude, period, and phase shift.
First, start plotting at the phase shift point \(x = -\frac{\pi}{6}\).

• This starting point corresponds to the horizontal shift due to the phase shift derived earlier.
• The function's vertical movement by +1 means adjusting your cosine baseline one unit upward.

Next, determine the end point of one complete period. Add the period \(\frac{2\pi}{3}\) to the starting x-coordinate:
\(x_{end} = -\frac{\pi}{6} + \frac{2\pi}{3}\), which concludes the period.
Within this range, carefully plot critical points like maximum, minimum, and intercepts to form the full wave cycle according to the calculated amplitude and shifted direction.