Problem 22 - Technical Mathematics with Calculus

Question

Graph each sine wave. Find the amplitude, period, and phase shift.$$y=4 \sin (4 x+\pi / 6)$$

Step-by-Step Solution

Answer

Amplitude: 4, Period: \(\frac{\pi}{2}\), Phase shift: \(-\frac{\pi}{24}\).

Step 1: Identify Amplitude

The amplitude of a sine function is the coefficient of the sine term. In the given function, the amplitude is given by the coefficient of the sine term, which is 4. This represents the maximum value the sine wave reaches above and below its midline.

Step 2: Determine the Period

The period of the sine function is how long it takes for the function to repeat its pattern. In a standard sine function that is in the form of \(y = \sin(Bx)\), the period is \(\frac{2\pi}{|B|}\). In the given function, \(B = 4\), so the period is \(\frac{2\pi}{4} = \frac{\pi}{2}\).

Step 3: Find Phase Shift

The phase shift of the function is determined by the horizontal translation of the sine function. The phase shift is calculated from the inside argument of the sine function, \(Bx + C\), where the phase shift is \(-\frac{C}{B}\). In the given function, \(C = \frac{\pi}{6}\) and \(B = 4\), so the phase shift is \(-\frac{\pi/6}{4} = -\frac{\pi}{24}\).

Step 4: Graph the Sine Wave

Using the amplitude, period, and phase shift, we can plot the sine wave on a coordinate plane. The sine wave will have a maximum value at 4 and a minimum value at -4. It completes one full cycle in \(\frac{\pi}{2}\) units along the x-axis, and it begins \(\frac{\pi}{24}\) units to the right of the y-axis.

Key Concepts

Amplitude of Sine Function

The amplitude of a sine function is a measure of its maximum vertical displacement from its midline. Essentially, it tells us how 'tall' the wave is. When looking at the sine function in the form of y = A sin(Bx + C), the coefficient A represents the amplitude. An easy way to remember this is that amplitude affects altitude. In our exercise, the given function is y = 4 sin(4x + \(\frac{\pi}{6}\)). Here, the amplitude is 4, indicating that the wave peaks at 4 units above the midline and troughs at 4 units below it.

For students: To properly identify the amplitude, always look for the absolute value of the coefficient in front of the sine term; negative values do not affect the amplitude as it's always a positive number, representing distance. Also, the amplitude does not change where the wave starts or ends; it only stretches or shrinks the wave vertically.

Period of Sine Function

The period of a sine function refers to the length it takes for the function to complete a full cycle before repeating itself. This concept is crucial in understanding the wave-like behavior of the sine function. It is mathematically defined for a standard sine function y = sin(Bx) as Period = \(\frac{2\pi}{|B|}\), where B is the frequency variable. Higher values of B compress the wave horizontally, resulting in a shorter period, while lower values of B stretch it out.

In our step-by-step solution from the exercise, the period is calculated using the formula with B = 4, giving us a period of \(\frac{\pi}{2}\). This tells us that every \(\frac{\pi}{2}\) units along the x-axis, the sine function starts repeating. Remember, changes in the period will impact how often the wave cycles but it doesn't shift the wave left or right along the x-axis—that's where the phase shift comes into play.

Phase Shift of Sine Function

Phase shift in a sine wave denotes a horizontal shift, moving the wave left or right on the coordinate plane. If you think of a sine wave as starting at the origin, the phase shift tells you how far from the origin the wave begins. It's identified by transforming the standard sine function into y = sin(Bx + C) and calculating the shift with the formula Phase shift = -\(\frac{C}{B}\). A positive shift moves the graph to the right, while a negative shift moves it to the left.

Looking back at our original function y = 4 sin(4x + \(\frac{\pi}{6}\)), we see that B is 4 and C is \(\frac{\pi}{6}\). Plugging these into our formula yields a phase shift of -\(\frac{\pi}{24}\). This negative shift means our sine wave will start to the left of the y-axis by \(\frac{\pi}{24}\) units. It's a slight leftward nudge on the graph from the typical starting point, and it's vital for accurately graphing the sine function and understanding its complex behavior in real-world applications like signal processing.