Problem 18 - Algebra 2

Question

State the amplitude, period, and phase shift for each function. Then graph the function. $$ y=\sin \left(\theta-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

Amplitude: 1; Period: \(2\pi\); Phase Shift: \(\frac{\pi}{4}\) to the right.

Step 1: Identify the Amplitude of the Function

The amplitude of a sine function, such as \( y = a \sin(b(\theta - c)) \), is given by the absolute value of the coefficient \( a \). In this function, \( y = \sin\left(\theta - \frac{\pi}{4}\right) \), the amplitude is 1 because there is no coefficient in front of the sine function other than 1.

Step 2: Determine the Period of the Function

The period of a sine function is calculated as \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( \theta \) inside the sine function. Here, \( b = 1 \), so the period is \( \frac{2\pi}{1} = 2\pi \).

Step 3: Calculate the Phase Shift of the Function

The phase shift of a sine function is determined by the formula \( \frac{c}{b} \), where \( c \) is the horizontal shift term added or subtracted inside the function, and \( b \) is the coefficient of \( \theta \). For this function, \( c = \frac{\pi}{4} \) and \( b = 1 \), thus the phase shift \( \frac{\pi}{4} \). The phase shift is to the right, since it is subtracted.

Step 4: Graph the Function

To graph \( y = \sin\left(\theta - \frac{\pi}{4}\right) \), start by plotting the standard sine curve, but shifted to the right by \( \frac{\pi}{4} \). The amplitude remains 1, so the graph oscillates between 1 and -1. The period remains \( 2\pi \), meaning it completes one full cycle every \( 2\pi \) units along the \( \theta \)-axis.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions. It is often written as \( y = \sin(\theta) \), where \( \theta \) is the angle measured in radians. This function is used to model periodic phenomena, such as waves or oscillations.

In its simplest form, the sine function has a repeating pattern over a domain from \( 0 \) to \( 2\pi \). During this interval, the function starts at 0, reaches its maximum at \( \frac{\pi}{2} \), returns to 0 at \( \pi \), reaches its minimum at \( \frac{3\pi}{2} \), and finishes the cycle back at 0 at \( 2\pi \). This results in a smooth, wave-like curve that is symmetric around the origin.

The sine function is used to understand and describe various properties of waves and oscillations, including their height, frequency, and movement in space.

Amplitude

The amplitude of a sine function measures the height of its waves. Specifically, it is the distance from the middle, or equilibrium point, to the peak of the wave. In equation form, it's represented as \( |a| \), where \( a \) is the coefficient in front of the sine function.

For the function \( y = \sin(\theta - \frac{\pi}{4}) \), the amplitude is 1, because there is no coefficient altering the sine function. Consequently, the graph oscillates between 1 and -1, which reflects a typical amplitude for a standard sine function.

• If \( a \) were greater than 1, the graph would stretch vertically, making the peaks higher.
• If \( a \) were less than 1 but greater than 0, the graph would compress vertically, making the peaks lower.
• A negative \( a \) would invert the wave.

The concept of amplitude is essential when dealing with real-world waveform problems, like sound and light waves.

Period

The period of a sine function is the horizontal length of one complete cycle of the wave. It is determined by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( \theta \) in the function. This measures how quickly the wave oscillates.

For \( y = \sin(\theta - \frac{\pi}{4}) \), the period remains \( 2\pi \), as there is no multiplying factor in front of \( \theta \) that alters its natural cycle. This means that after \( 2\pi \) units along the \( \theta \)-axis, the sine wave repeats its pattern.

• A larger \( b \) would lead to a shorter period, causing the wave to compress horizontally, oscillating more frequently.
• A smaller \( b \) would result in a longer period, elongating the wave and slowing the oscillation.

Understanding the period is critical in applications like signal processing and harmonic analysis, where the frequency and timing of waves are crucial.

Phase Shift

Phase shift refers to the horizontal translation of a trigonometric function on the graph. It tells us how far the function is shifted from its original position along the \( \theta \) axis. The phase shift is calculated using the formula \( \frac{c}{b} \), where \( c \) is the horizontal shift value, and \( b \) is the coefficient of \( \theta \) in the function.

In the case of \( y = \sin(\theta - \frac{\pi}{4}) \), the phase shift is \( \frac{\pi}{4} \), moving the graph to the right. This is because the function is written as \( \theta - \frac{\pi}{4} \), indicating a positive phase shift if we think of it as \( \theta - c \).

Phase shifts are useful in synchronizing waves and understanding time-dependent behaviors in a variety of fields:

• In audio engineering, adjusting phase shifts can help to better align the waves to reduce destructive interference.
• In engineering, phase shifts can be crucial for timing circuits and wave modulation.

Understanding how a phase shift affects a graph helps learners visually and conceptually grasp the behavior and movement of sine waves.