Problem 19 - Algebra 2

Question

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\cos \left(\theta+\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the standard form of the cosine function

The standard form of the cosine function is \( y = a \cos(b\theta + c) + d \), where \( a \) is the amplitude, \( b \) helps determine the period, \( c \) affects the phase shift, and \( d \) is the vertical shift. Here, \( y = \cos \left(\theta + \frac{\pi}{3}\right) \).

Step 2: Find the amplitude

The amplitude of a cosine function given by \( y = a \cos(b\theta + c) + d \) is the absolute value of \( a \). In this case, \( a = 1 \), so the amplitude is \( |1| = 1 \).

Step 3: Calculate the period

The period of the cosine function is given by \( \frac{2\pi}{|b|} \). Here, the value of \( b = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).

Step 4: Determine the phase shift

The phase shift of a cosine function \( y = a \cos(b\theta + c) + d \) is calculated by \( -\frac{c}{b} \). Here, \( c = \frac{\pi}{3} \) and \( b = 1 \). Therefore, the phase shift is \( -\frac{\pi}{3} \).

Step 5: Graph the function

To graph the function \( y = \cos \left(\theta + \frac{\pi}{3}\right) \), start with the basic graph of \( y = \cos(\theta) \). Shift the entire graph to the left by \( \frac{\pi}{3} \) units, keeping the amplitude and period unchanged.

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often represented as \( y = \cos(\theta) \). It describes the relationship between the angle \( \theta \) and the x-coordinate of a point on the unit circle. A key feature of the cosine curve is its wave-like shape, which repeats at regular intervals. This cyclic repetition is what makes the cosine function periodic.

• The standard form of a cosine function can be expressed as: \( y = a \cos(b\theta + c) + d \)
• Here, \( a \), \( b \), \( c \), and \( d \) are constants that modify the function's characteristics such as amplitude, period, and phase shift.

Understanding these constants is essential for manipulating and graphing cosine functions effectively.

Amplitude

Amplitude refers to the maximum height or distance from the center line (horizontal axis) to the peak or trough of the wave. It provides a measure of how "tall" or "short" the wave is.

• In the equation \( y = a \cos(b\theta + c) + d \), the amplitude is \( |a| \).
• For the given function \( y = \cos \left(\theta + \frac{\pi}{3}\right) \), the amplitude is \( |1| = 1 \).

The amplitude indicates that the wave reaches a maximum height of 1 unit above and below the center line (y=0). This means the wave oscillates equally in both directions from the center.

Period

The period of a cosine function defines how long it takes for the wave pattern to complete one full cycle. Essentially, it's the length required for the function to return to its starting point.

• For the cosine function \( y = a \cos(b\theta + c) + d \), the period is calculated using \( \frac{2\pi}{|b|} \).
• In the function given, \( b = 1 \), leading to a period of \( \frac{2\pi}{1} = 2\pi \).

A period of \( 2\pi \) tells us the wave pattern repeats every \( 2\pi \) units along the x-axis, maintaining the same shape and size throughout.

Phase Shift

Phase shift is the horizontal movement of the graph of the function along the x-axis, determining where the wave begins along the input axis.

• It is calculated using the formula \( -\frac{c}{b} \) for the function \( y = a \cos(b\theta + c) + d \).
• In our function, \( c = \frac{\pi}{3} \) and \( b = 1 \), resulting in a phase shift of \( -\frac{\pi}{3} \).

This means the entire cosine wave is shifted \( \frac{\pi}{3} \) units to the left. This significant characteristic allows us to map out where the wave starts compared to the standard cosine function \( y = \cos(\theta) \). Understanding phase shifts helps in correctly plotting the wave's starting position on a graph.