Question

Graph each sine wave. Find the amplitude, period, and phase shift.$$y=\sin 3 x$$

Short answer

Amplitude is 1, period is \(\frac{2\pi}{3}\), and phase shift is 0. Graph using points at each quarter interval of the period.

Step-by-step solution

Identify the Amplitude

The amplitude of a sine function is the coefficient of the sine term when the equation is in the form of \(y = A \sin(B(x - C)) + D\), where A is the amplitude. For the equation \(y = \sin 3x\), the coefficient of the sine term is 1, which means the amplitude is 1.

Determine the Period

The period of a sine function is calculated by \(P = \frac{2\pi}{|B|}\), where B is the coefficient of \(x\) inside the sine function. For the equation \(y = \sin 3x\), \(B = 3\), so the period is \(P = \frac{2\pi}{3}\).

Calculate the Phase Shift

The phase shift of a sine function in the form \(y = A \sin(B(x - C)) + D\) is given by C. If the term \((x - C)\) is not present, the phase shift is 0. In \(y = \sin 3x\), there is no phase shift mentioned, so the phase shift is 0.

Sketch the Graph

To graph \(y = \sin 3x\), plot key points over one period from \(0\) to \(\frac{2\pi}{3}\) based on the sine function's known shape. Plot points at \(x = 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2\pi}{3}\). Notice that the wave repeats every \(\frac{2\pi}{3}\) units along the x-axis and extends from \(1\) to \(-1\) on the y-axis due to the amplitude of 1.

Key concepts

Amplitude of Sine Function

The amplitude of a sine function represents the height of its peaks and the depth of its troughs from the center line of the graph. In simple terms, amplitude determines how tall or short the waves are.

For any sine function in the standard form of \(y = A\sin(B(x - C)) + D\), the amplitude is represented by the absolute value of \(A\). If \(A\) is positive, the sine wave starts at the center line and moves upwards. If \(A\) is negative, it indicates that the sine wave is reflected across the horizontal axis.

For our specific exercise, the equation \(y = \sin 3x\) has an implied coefficient of 1 in front of the sine term. Therefore, the amplitude of this sine wave is 1. This means that the maximum and minimum values the function can reach on the graph are 1 and -1, respectively. A solid understanding of amplitude is vital for correctly sketching the vertical stretch of wave patterns on a coordinate grid.

Period of Sine Function

The period of a sine function is the distance along the x-axis required for the wave to complete one full cycle and begin to repeat itself. It dictates how 'squished' or 'stretched' the sine wave appears horizontally.

In mathematical notation, the period \(P\) is found using the formula \(P = \frac{2\pi}{|B|}\), where \(B\) is the coefficient of \(x\) in the sine function, as shown in the general form \(y = A\sin(B(x - C)) + D\).

In the exercise's equation \(y = \sin 3x\), the period is calculated as \(\frac{2\pi}{3}\) because \(B = 3\). This means that every \(\frac{2\pi}{3}\) units, the sine wave pattern repeats itself. Students should practice determining the period of different sine functions, as this greatly aids in understanding their horizontal expansion or compression, a necessary step for accurate graphing.

Phase Shift of Sine Function

Phase shift in a sine function refers to the horizontal displacement of the wave pattern along the x-axis. In other words, if a phase shift is present, the entire graph of the sine wave moves to the right or left.

This shift is determined by the \(C\) value in the formula \(y = A\sin(B(x - C)) + D\), signifying that the graph is shifted \(C\) units to the right if \(C\) is positive, or \(C\) units to the left if \(C\) is negative.

When the equation is in the standard form and lacks a \((x - C)\) term, like in our exercise's equation \(y = \sin 3x\), the phase shift is 0. This indicates that there is no horizontal movement of the wave pattern, and the sine function starts its cycle at \(x = 0\). Recognizing the absence or presence of a phase shift is essential when plotting the starting point of a sine wave on a graph.