Problem 5 - Applied Calculus

Question

Sketch graphs of the functions. What are their amplitudes and periods? $$y=3 \sin 2 x$$

Step-by-Step Solution

Answer

Amplitude: 3; Period: \(\pi\).

Step 1: Identify the Standard Sine Function

The standard sine function is given by the equation \(y = a \sin(bx)\), where \(a\) is the amplitude and \(\frac{2\pi}{b}\) is the period of the function. In the given function, \(y = 3 \sin 2x\), here \(a = 3\) and \(b = 2\).

Step 2: Determine the Amplitude

The amplitude of a sine function is the coefficient \(a\) in front of the sine function. It represents the maximum value that the sine function reaches. For the given function \(y = 3 \sin 2x\), this coefficient is \(3\). Therefore, the amplitude is \(3\).

Step 3: Calculate the Period

The period of the function is determined by the coefficient \(b\). The period is calculated using the formula \(\frac{2\pi}{b}\). Substituting \(b = 2\) from the given function \(y = 3 \sin 2x\), we get:\[\text{Period} = \frac{2\pi}{2} = \pi\]So, the period of the function is \(\pi\).

Step 4: Sketch the Graph

To sketch the graph of the function \(y = 3 \sin 2x\), start by noting that it is a sine curve that oscillates between \(-3\) and \(3\), with a period of \(\pi\). This means one full wave of the sine function fits within an interval of \(\pi\) on the \(x\)-axis. Plot key points at \(x = 0\), \(x = \frac{\pi}{2}\), \(x = \pi\), \(x = \frac{3\pi}{2}\), and \(x = 2\pi\), noting the function's value at these points.

Key Concepts

Sine Function Graphing

The sine function is a fundamental function in trigonometry, notable for its wave-like shape. When you graph a standard sine function, such as \( y = \sin x \), it will oscillate smoothly between 1 and -1. To graph any sine function, you need to identify a few key characteristics, which include the amplitude, period, and vertical/horizontal shifts, if present. For instance, in the function \( y = 3 \sin 2x \), these elements directly influence the shape and position of the sine curve on the graph.
To graph the function, you should:

• Identify the amplitude, which dictates the height of the wave.
• Determine the period, which tells you how long it takes for the function to complete one full cycle on the x-axis.
• Check for any phase shifts or vertical translations.

With this function, \( y = 3 \sin 2x \) graphs as a regular sine wave that is stretched or compressed vertically and horizontally based on the amplitude and period. It achieves its maximum and minimum values within the bounds determined by the amplitude, all within the periodic intervals determined by the period.

Amplitude in Trigonometry

Amplitude is a key concept when dealing with sine functions. It is essentially the measure of how far the function's values reach from its central axis, acting as a "scale factor" vertically for the sine wave. In the function \( y = 3 \sin 2x \), the amplitude is \(3\). This means the graph of the function reaches up to \(3\) and down to \(-3\) around the central horizontal axis (which is often the x-axis for unshifted functions).

Amplitude affects the "height" of the wave:

• It does not change the wave's shape or period, only how tall the peaks and troughs are.
• The greater the amplitude, the larger the peak-to-trough distance.
• A magnitude of \(a = 0\) would result in a flat line, as there would be no vertical distance to measure.

Understanding amplitude helps in quickly determining how "intense" or "damped" a sine wave appears, describing the extent of its oscillations above and below the horizontal axis.

Periodicity in Trigonometric Functions

Periodicity refers to the repeating nature of trigonometric functions. The period of a sine function determines how frequently the wave pattern repeats itself over the x-axis. For the function \( y = 3 \sin 2x \), the period is determined by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) in the sine function. Here, \( b = 2 \), leading to a period of \( \pi \).

Key points about the period:

• A period of \( \pi \) means the sine function completes one full wave cycle within an interval of length \( \pi \).
• The periodic nature of sine functions means they repeat the same set of values within these cycles.
• By determining the period, you know the "width" of each wave cycle, which is crucial for graphing multiple cycles of the function.

Understanding periodicity is essential for analyzing how sine functions behave over different intervals, allowing for accurate graphing and application across various mathematical and real-world contexts.