Problem 21 - Algebra 2 and Trigonometry

Question

Find the phase shift of each function. \(y=\sin \left(x+\frac{\pi}{3}\right)\)

Step-by-Step Solution

Answer

The phase shift is \( \frac{\pi}{3} \) to the left.

Step 1: Understanding the Standard Form

The standard form of the sine function with a phase shift is given by:\[ y = ext{a} imes ext{sin}( ext{b}x - ext{c}) + d \]where the phase shift is expressed as \( -\frac{c}{b} \). Compare the given function with this standard form to identify the values of \(b\) and \(c\).

Step 2: Identifying Values from the Function

In the given function \( y = \sin \left( x + \frac{\pi}{3} \right) \), we can rewrite it as:\[ y = \sin \left( x - \left( -\frac{\pi}{3} \right) \right) \]Comparing with the standard form, we find:- \( b = 1 \)- \( c = -\frac{\pi}{3} \)

Step 3: Calculating the Phase Shift

The phase shift formula is \( -\frac{c}{b} \). Using the values identified:\[ -\frac{c}{b} = -\left( \frac{-\frac{\pi}{3}}{1} \right) = \frac{\pi}{3} \]Thus, the phase shift is \( \frac{\pi}{3} \) to the left.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry, representing a periodic wave. Its graph oscillates between 1 and -1, repeating every \( 2\pi \) radians. This sine wave is smooth and continuous.
The basic sine function is written as \( y = \sin(x) \), where \( x \) represents the angle in radians. The function outputs the sine of this angle. The peak of the sine wave occurs at \( \frac{\pi}{2} \), \( \frac{5\pi}{2} \), etc., while the troughs occur at \( \frac{3\pi}{2} \), \( \frac{7\pi}{2} \), and so on.
**Key Characteristics:**

• Amplitude: The height from the middle to the peak is always 1 in the basic form.
• Period: The wave repeats every \( 2\pi \) radians.
• Midline: The average value of the wave is 0, lying on the x-axis in the simplest form.

These characteristics make the sine function crucial for modeling real-world periodic phenomena, such as sound waves or tidal patterns.

Phase Shift Formula

The phase shift formula helps us understand how much a trigonometric function is shifted horizontally from its usual position. For sine and cosine functions, the formula is applied as part of the standard form equation.
The phase shift of a trigonometric function \ can be calculated using the formula \( -\frac{c}{b} \). Here, **c** is the horizontal shift, and **b** affects the function's period. This formula tells us the amount and direction of the shift along the x-axis.
In our specific example, we have the function \( y = \sin \left( x + \frac{\pi}{3} \right) \). When rewritten, it becomes \( y = \sin \left( x - \left( -\frac{\pi}{3} \right) \right) \). By identifying \( b = 1 \) and \( c = -\frac{\pi}{3} \), we plug these into the phase shift formula:\[-\frac{c}{b} = -\left( \frac{-\frac{\pi}{3}}{1} \right) = \frac{\pi}{3}.\]This gives us a phase shift of \( \frac{\pi}{3} \), indicating the graph shifts \( \frac{\pi}{3} \) units to the left. Understanding phase shifts is key to correctly graphing and interpreting these functions.

Standard Form of Trigonometric Functions

To analyze and graph trigonometric functions effectively, it's essential to understand their standard form. The standard form for sine and cosine functions is: \[ y = a \sin(bx - c) + d \] This format includes parameters that describe transformations of the basic wave:

• a indicates amplitude changes, affecting the wave’s height.
• b changes the period of the wave, calculated by \( \frac{2\pi}{b} \).
• c contributes to the phase shift, moving the wave horizontally.
• d shifts the wave vertically, altering the midline.

By comparing a given trigonometric function to this standard form, one can identify these parameters and understand how the graph is transformed from the basic sine or cosine wave. In our example function \( y = \sin(x + \frac{\pi}{3}) \), \( a = 1 \), \( b = 1 \), \( c = -\frac{\pi}{3} \), and \( d = 0 \). Understanding this form helps in predicting and visualizing changes in the wave's behavior effectively.