Problem 9 - Precalculus with Limits

Question

Finding the Period and Amplitude, find the period and amplitude. $$ y=\frac{1}{2} \sin \frac{\pi x}{3} $$

Step-by-Step Solution

Answer

The amplitude of the function \( y=\frac{1}{2} \sin \frac{\pi x}{3} \) is \( \frac{1}{2} \) and the period is \( 6 \).

Step 1: Identify the Amplitude

The amplitude is the coefficient of the sine function. In this case, the amplitude is given by \( \frac{1}{2} \). So the amplitude of \( y=\frac{1}{2} \sin \frac{\pi x}{3} \) is \( \frac{1}{2} \).

Step 2: Identify the Period

The period of a sine function is given by \( \frac{2\pi}{\text{coefficient of } x} \). Here the coefficient of \( x \) is \( \frac{\pi}{3} \), so the period is \( \frac{2\pi}{\frac{\pi}{3}} = 2 \cdot 3 = 6 \). So the period of \( y=\frac{1}{2} \sin \frac{\pi x}{3} \) is \( 6 \).

Key Concepts

Amplitude

Amplitude is a crucial concept when dealing with trigonometric functions like sine and cosine. It tells us how high and how low the graph of the function will go relative to its midline. In the sine function represented by the equation \( y = \frac{1}{2} \sin \frac{\pi x}{3} \), the amplitude can be identified as the value multiplying the sine function, which is \( \frac{1}{2} \). This means the function will oscillate from \( +\frac{1}{2} \) to \( -\frac{1}{2} \) around its horizontal axis.
Amplitude has the following characteristics:

• It only affects the vertical stretching or compressing of the graph, not the sideways behavior.
• Amplitude is always a positive value, even if the coefficient is negative; the sign indicates direction, not height.
• A smaller amplitude results in a flatter wave, whereas a larger amplitude results in a taller wave.

Understanding amplitude helps predict how dramatic the vertical changes will be in a sine wave.

Period

The period of a trigonometric function like a sine wave tells us how long it takes for the graph to complete one full cycle. For our function \( y = \frac{1}{2} \sin \frac{\pi x}{3} \), calculating the period involves identifying the coefficient of \( x \) inside the sine function.
In this case, the coefficient is \( \frac{\pi}{3} \). To calculate the period, use the formula for the sine function's period: \( \frac{2\pi}{\text{coefficient of } x} \). Plugging in the values gives:
\[ \frac{2\pi}{\frac{\pi}{3}} = 2 \cdot 3 = 6 \]
Thus, the period of this function is 6.

• A higher coefficient of \( x \) results in a shorter period, meaning more cycles in a given interval.
• A lower coefficient results in a longer period, with the cycles stretched out over more space.
• Understanding the period allows you to know how frequently a sine wave repeats itself.

Recognizing the period is essential for predicting the horizontal repetition of the wave.

Sine Function

The sine function is one of the core trigonometric functions, fundamentally important in mathematics and many real-world applications like physics and engineering. The standard sine function, \( y = \sin x \), produces a smooth periodic wave which repeats every \( 2\pi \) units.
In our function \( y = \frac{1}{2} \sin \frac{\pi x}{3} \), the sine function undergoes two key transformations:
1. **Amplitude Change:** By multiplying by \( \frac{1}{2} \), the wave's height is halved.
2. **Period Change:** The coefficient \( \frac{\pi}{3} \) inside the function modifies its period, as calculated, to 6.
Sine functions have several important properties:

• They are smooth, continuous curves that oscillate between -1 and 1 in their standard form.
• The waves are symmetric about their midline, which can be shifted by vertical translations.
• Sine functions model phenomena that repeat at regular intervals, such as sound waves or tides.

Grasping how the sine function works, and how adjustments affect its graph, is pivotal for understanding its behavior across different scenarios.