Problem 40 - Precalculus with Calculus Previews

Question

Find the amplitude, period, and phase shift of the given function. Sketch at least one cycle of the graph. $$ y=\sin \left(3 x-\frac{\pi}{4}\right) $$

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(\frac{2\pi}{3}\), Phase shift: \(\frac{\pi}{12}\) to the right.

Step 1: Identify Function Parameters

First, identify the parameters A, B, C, and D from the function of the form \(y = A\sin(Bx - C) + D\). In the given function, \(y = \sin(3x - \frac{\pi}{4})\), we have \(A = 1\), \(B = 3\), \(C = \frac{\pi}{4}\), and \(D = 0\).

Step 2: Calculate Amplitude

The amplitude of a sine function is the absolute value of A. Here, \(A = 1\), so the amplitude is \(1\).

Step 3: Calculate Period

The period of a sine function is calculated by \(\frac{2\pi}{B}\). Since \(B = 3\), the period is \(\frac{2\pi}{3}\).

Step 4: Determine Phase Shift

The phase shift of the sine function is calculated by \(\frac{C}{B}\). Given \(C = \frac{\pi}{4}\) and \(B = 3\), the phase shift is \(\frac{\pi}{12}\), and since it is subtracted, it is a shift to the right.

Step 5: Sketch the Graph

To sketch one cycle of the function, start from the phase shift \(x = \frac{\pi}{12}\), and sketch the wave from this point covering half the cycle until \(x = \frac{\pi}{12} + \frac{\pi}{3}\). Ensure you mark key points like maxima, minima, and zero crossings, noting the period and amplitude.

Key Concepts

Amplitude

The amplitude of a trigonometric function like the sine function represents how far the function's values stretch above and below its central axis. Think of it as the height of a wave.
For the function given, \(y = \sin(3x - \frac{\pi}{4})\), the amplitude is determined by the coefficient \(A\) in front of the sine function. Here, \(A = 1\), which means the highest point the wave reaches is 1 unit above the x-axis, and the lowest point is 1 unit below.
When the amplitude is 1, the wave retains its usual size and shape without any vertical stretch or compression. It's crucial to remember that the amplitude is always a positive number, representing the distance from the centerline (or the average value) of the sine wave to any of its peaks.

Period

The period of a trigonometric function defines how long it takes for the function to repeat its complete cycle. Imagine it as the length of one wave from one peak to the next.
For the sine function \(y = \sin(3x - \frac{\pi}{4})\), the period is calculated using the formula \(\frac{2\pi}{B}\), where \(B\) is the coefficient of \(x\). In this case, \(B = 3\), so the period is \(\frac{2\pi}{3}\).
This tells us that every \(\frac{2\pi}{3}\) units along the x-axis, the wave starts anew. The shorter period, compared to the usual \(2\pi\) of the basic sine function, means the waves are more frequent and compact.

Phase Shift

The phase shift of a trigonometric function reveals how the function is moved horizontally along the x-axis. It's akin to deciding where the wave starts its cycle.
In the function \(y = \sin(3x - \frac{\pi}{4})\), the phase shift can be calculated using \(\frac{C}{B}\). Here, \(C = \frac{\pi}{4}\) and \(B = 3\), resulting in a phase shift of \(\frac{\pi}{12}\).
Since the phase shift is positive and the equation is in the form \(Bx - C\), the wave is shifted to the right. This means the starting point of the wave on the graph has moved \(\frac{\pi}{12}\) units in a positive direction along the x-axis. Understanding phase shifts helps in accurately sketching or predicting the wave's behavior on a graph.