Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a two-period interval. State the phase shift. $$y=\sin \left(2 x+\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The phase shift is \(-\frac{\pi}{8}\). Graph the function from \(-\frac{\pi}{8}\) to \(\frac{15\pi}{8}\).

Step 1: Identify the Function

We have the function given as \(y=\sin \left(2 x+\frac{\pi}{4}\right)\). It's a sine function with a frequency change and a phase shift.

Step 2: Determine the Period

The formula for the period \( T \) of a sine function \( y = \sin(bx) \) is \( T = \frac{2\pi}{b} \). In our function \( b = 2 \), so the period is \( T = \frac{2\pi}{2} = \pi \). We need to graph this over a two-period interval, which is from 0 to \( 2\pi \).

Step 3: Identify the Phase Shift

The phase shift for a function \( y = \sin(bx + c) \) is given by \( \frac{-c}{b} \). For our function, \( c = \frac{\pi}{4} \) and \( b = 2 \). Thus, the phase shift is \( \frac{-\frac{\pi}{4}}{2} = -\frac{\pi}{8} \). This means the graph shifts to the left by \( \frac{\pi}{8} \).

Step 4: Set Up the Graph

To graph \( y = \sin\left(2x + \frac{\pi}{4}\right) \) over two periods, start by plotting the original sine wave \( y = \sin(2x) \) to go from its starting point of 0 to the end point at \( 2\pi \), adjusting all points by the phase shift \(-\frac{\pi}{8}\). Make sure key points such as the start, maximum, minimum, and end are appropriately marked.

Step 5: Plot Key Points of the Graph

For one period of the transformed sine function (\([0, \pi]\)), key points include:- Start: \( x = -\frac{\pi}{8} \), \( y = 0 \).- 1/4 Period \( \pi/4 \): \( x = \frac{\pi}{4} - \frac{\pi}{8} = \frac{\pi}{8} \), \( y = 1 \).- 1/2 Period \( \pi/2 \): \( x = \frac{\pi}{2} - \frac{\pi}{8} = \frac{3\pi}{8} \), \( y = 0 \).- 3/4 Period \( 3\pi/4 \): \( x = \frac{3\pi}{4} - \frac{\pi}{8} = \frac{5\pi}{8} \), \( y = -1 \).- End: \( x = \pi - \frac{\pi}{8} = \frac{7\pi}{8} \), \( y = 0 \). Extend these points accordingly to two periods.

Step 6: Draw the Graph

Using the points from Step 5, draw the sine wave over the interval \([-\frac{\pi}{8}, \frac{15\pi}{8}]\). Make sure the wave starts at the phase shift point and repeats its increasing and decreasing behavior two times in this range.

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions, often denoted as \( y = \sin(x) \). It describes a smooth wave that oscillates above and below the horizontal axis, providing a periodic pattern that repeats every \( 2\pi \) radians. This classic sinusoidal wave is characterized by its distinctive shape, which is symmetrical and continuous. Key features of the sine function include:

• Amplitude: It determines the height or strength of the wave. For the basic sine function, the amplitude is 1, which means the wave peaks at 1 and troughs at -1.
• Period: This defines the length of one complete cycle of the wave. For \( \sin(x) \), the period is \( 2\pi \).
• Midline: This is the horizontal line around which the wave oscillates. It's the average of the maximum and minimum values of the function, typically at \( y = 0 \) for the basic sine function.
• Points of intersection: The sine wave crosses the midline or x-axis at certain values, such as \( 0, \pi, 2\pi \), etc.

Understanding these properties helps in graphing transformations of the sine function, such as changes in amplitude, frequency, and phase shift.

Phase Shift

Phase shift refers to the horizontal movement of a trigonometric graph along the x-axis. In simple terms, it determines where the function starts its cycle. For a sine function of the form \( y = \sin(bx + c) \), the phase shift is calculated as \( \frac{-c}{b} \). How Phase Shift Works: - If the phase shift is negative, the graph shifts left.- If positive, it shifts right.In our example \( y = \sin(2x + \frac{\pi}{4}) \), the phase shift is \( \frac{-\frac{\pi}{4}}{2} = -\frac{\pi}{8} \). This means the sine wave starts \( \frac{\pi}{8} \) units to the left than it normally would. Such shifts are crucial for matching theoretical functions with real-world phenomena where the starting point might not coincide with \( x = 0 \). To visualize:- Identify the phase shift's direction and magnitude.- Adjust the key points of the sine wave accordingly, ensuring you account for this shift during graphing.

Period of Sine Function

The period of a sine function defines the length of one complete cycle of the wave. In the function form \( y = \sin(bx) \), the period \( T \) is calculated using the formula \( T = \frac{2\pi}{b} \).Importance of the Period:

• It defines the frequency of oscillations or how often the cycle repeats over an interval.
• A shorter period indicates more frequent oscillations, while a longer period means the wave stretches out more vertically.

For the function \( y = \sin(2x + \frac{\pi}{4}) \), we have \( b = 2 \). Thus, the period is \( \frac{2\pi}{2} = \pi \). This implies the wave completes a full cycle as \( x \) varies from 0 to \( \pi \), rather than the standard \( 2\pi \) for the basic sine function.By understanding the period:- You can predict the frequency of oscillations.- When graphing, determine the x-values that capture exactly one or more complete cycles, which is helpful when visualizing the function over multiple periods.[Add example of one period]