Problem 41 - Algebra and Trigonometry

Question

\(29-42\) . Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=\sin (\pi+3 x) $$

Step-by-Step Solution

Answer

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

Step 1: Rewrite the Function

The given function is \( y = \sin(\pi + 3x) \). Rewrite it to match the form \( y = \sin(Bx + C) \). The function is already in this form with \(B = 3\) and \(C = \pi\).

Step 2: Find the Amplitude

The amplitude of the function \( y = \sin(Bx + C) \) is 1, because the coefficient in front of the sine function is 1.

Step 3: Calculate the Period

The period of the sine function is calculated as \( \frac{2\pi}{|B|} \). Substitute \(B = 3\), so \( \text{Period} = \frac{2\pi}{3} \).

Step 4: Determine the Phase Shift

The phase shift is calculated as \( -\frac{C}{B} \). Here, \(C = \pi\) and \(B = 3\), giving the phase shift as \( -\frac{\pi}{3} \).

Step 5: Graph One Complete Period

To graph one complete period, start at the phase shift, which is \(-\frac{\pi}{3}\), and complete the cycle at \( -\frac{\pi}{3} + \frac{2\pi}{3} \) which is \( \frac{\pi}{3} \). Plot the sine curve starting at this phase shift over the computed period, keeping the amplitude at 1.

Key Concepts

Amplitude

In trigonometric functions, the amplitude is a measure of how "tall" or "big" the wave is, meaning how far it deviates from its central axis. It defines the maximum extent of a wave measured from its equilibrium (rest) position. The amplitude affects the vertical stretch of the sine or cosine functions.For the function provided, \[y = \sin(\pi + 3x) \],the coefficient of the sine function (which is the number before the sine term) defines the amplitude. In this case, there's no numerical coefficient explicitly written before the sine term, which means it defaults to 1. Thus, this function has an amplitude of 1. This means the wave will oscillate from 1 to -1 around the origin. If there were a number other than 1 present, such as 2, then the wave would reach as high as 2 and as low as -2. Understanding amplitude is crucial because it affects how we view the nature and behavior of wave-like phenomena exhibited by trigonometric functions.

Period

The period of a trigonometric function indicates how quickly it repeats its cycle. More formally, it's the distance along the x-axis that it takes before the function starts to repeat itself. For sine and cosine functions, the usual period is \(2\pi \),indicating one full cycle around the unit circle.In our function, \[y = \sin(\pi + 3x)\], the formula to find the period of a sine function is \(\frac{2\pi}{|B|}\),where \(B\) is the coefficient of \(x\). Here, \(B\) equals 3, giving us a period of \(\frac{2\pi}{3} \). This tells us the sine wave completes one full cycle from peak to peak over an interval of \(\frac{2\pi}{3}\).Understanding the period is essential for graphing trigonometric functions accurately and for predicting the way these functions behave.Knowing that this function has a shortened period compared to the standard sine function indicates that the wave oscillates more frequently along the x-axis.

Phase Shift

Phase shift refers to the horizontal movement of a trigonometric function along the x-axis. It reveals how far and in what direction (left or right) a graph is shifted from its usual position. For a function like sine or cosine, this shift can drastically change the graph's starting point.For the function \[y = \sin(\pi + 3x)\], the phase shift is determined using the formula \(-\frac{C}{B}\),where \(B\) is 3, and \(C\) is \(\pi\). By substituting the values into the formula, the phase shift computes as \(-\frac{\pi}{3}\). This negative phase shift indicates the sine wave will start \(\frac{\pi}{3}\)units to the left of where it would typically begin. Understanding the phase shift helps in plotting the function accurately, as it tells you exactly where the wave commences relative to the usual starting point found at the origin in the absence of any phase shift.Properly accounting for phase shifts is essential in many applications of trigonometry in sciences and engineering, where aligning waveforms accurately can be crucial.