Problem 24 - Precalculus: Functions and Graphs

Question

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=4 \sin 3 \pi x\)

Step-by-Step Solution

Answer

Amplitude is 4, period is \(2/3\), no phase shift.

Step 1: Identify the amplitude

The amplitude of a sinusoidal function like \(y = a \sin(bx + c) + d\) is the absolute value of the coefficient in front of the sine function. For the function \(y = 4 \sin(3\pi x)\), the amplitude is \(4\).

Step 2: Calculate the period

The period of the function \(y = a \sin(bx + c) + d\) is given by \(\frac{2\pi}{b}\). For the function \(y = 4 \sin(3\pi x)\), \(b = 3\pi\). Therefore, the period is \(\frac{2\pi}{3\pi} = \frac{2}{3}\).

Step 3: Determine the phase shift

The phase shift of the function \(y = a \sin(bx + c) + d\) is \(-\frac{c}{b}\). Here, since there is no \(c\) term in \(y = 4 \sin(3\pi x)\), the phase shift is \(0\).

Step 4: Sketch the graph

To sketch the graph of \(y = 4\sin(3\pi x)\), start by drawing the sine curve with a maximum amplitude of 4 and a minimum amplitude of -4. The curve completes one full cycle every \(\frac{2}{3}\) units along the x-axis. The graph has no phase shift, so it starts at \(x = 0\). Draw the curve by marking the key points at \(x = 0\), \(x = \frac{1}{6}\), \(x = \frac{1}{3}\), \(x = \frac{1}{2}\), and continue repeating the pattern.

Key Concepts

Amplitude

In any sinusoidal function, the amplitude is a critical factor that helps define its shape. Think of it as the height of the waves. Mathematically, the amplitude is the absolute value of the coefficient in front of the sine or cosine function. In our case, we have the function \(y = 4 \sin(3\pi x)\). Here, the coefficient of the sine function is \(4\). This tells us that the amplitude is \(4\), which indicates the maximum distance from the midline to the peak of the wave.

When you look at a graph drawn for this function, the curve will rise up to \(4\) and dip down to \(-4\).
Remember, the amplitude is always a positive number because it simply measures the distance, without concern for direction.

Period

The period of a sinusoidal function shows how often the function repeats its cycle. It's the length it takes to complete one full wave cycle on the graph. For any function \(y = a \sin(bx + c) + d\), we calculate the period using the equation \(\frac{2\pi}{b}\).

In \(y = 4 \sin(3\pi x)\), \(b\) is \(3\pi\). Plugging this into our formula, the period is \[\frac{2\pi}{3\pi} = \frac{2}{3}\].

So, every \(\frac{2}{3}\) units along the x-axis, you'll witness the entire wave starting a fresh cycle. This is shorter than the standard sine wave's period of \(2\pi\), showing how compressed the waves are due to a larger \(b\) value.

Phase Shift

Phase shift refers to how the function shifts horizontally along the x-axis compared to a regular sine function \(y = \sin(x)\). For the function \(y = a \sin(bx + c) + d\), the phase shift is determined by the formula \(-\frac{c}{b}\).

In our example \(y = 4 \sin(3\pi x)\), there is no \(c\) or horizontal translation term. This translates to a \(c\) of \(0\), resulting in no horizontal shift. Therefore, the phase shift is \(0\).
This means that the wave starts precisely at the origin \(x = 0\), ensuring a straightforward pattern without any left or right shifts.

The understanding of phase shift is essential for accurately plotting and predicting the intricacies of sinusoidal graphs.