Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=-2 \sin \frac{2 \pi x}{3}$$

Step-by-Step Solution

Answer

The amplitude of the function is 2 and the period is 3. The graph is a reflection of the standard sine function over the x-axis with a stretch by a factor of 2 in the y-direction and a stretch by a factor of 3 in the x-direction.

Step 1: Determining the Amplitude

The amplitude of the function is given by the absolute value of the coefficient of the sine function. Thus, the amplitude is \( |-2| = 2 \).

Step 2: Determining the Period

The period of a sine function is given by \( \frac{2 \pi}{|B|} \) where B is the coefficient of x inside the sine function. Here, the coefficient of x is \( \frac{2 \pi}{3} \), thus the period is \( \frac{2 \pi}{\frac{2 \pi}{3}} = 3 \).

Step 3: Graphing the Function

Graph the function \( y = -2 \sin \frac{2 \pi x}{3} \) taking into account the amplitude, the period and by remembering that the graph should be reflected about the x-axis because of the negative coefficient in front of the sine function. Make sure to include at least two full periods of the function on the graph.

Key Concepts

Amplitude

Amplitude is an essential characteristic for sine functions. It measures the maximum distance of the curve from the horizontal axis, which is typically the x-axis in a graph. For the sine function, amplitude is determined by

• Finding the coefficient in front of the sine function.
• Taking the absolute value of this coefficient.

This gives the amplitude equation: \[|A|\]where \( A \) is the coefficient.
In the function \( y = -2 \sin \frac{2 \pi x}{3} \), the coefficient is \(-2\). Thus, its amplitude is \( | -2 | = 2 \).

Amplitude is always a positive value, showing how far up and down the graph reaches from its midline. In simpler terms, if you measure from the center up to the highest point or down to the lowest point, you encounter the same number, known as the amplitude.

Period of a Function

The period of a sine function indicates the length over which the function starts repeating itself. A full cycle goes from one point, completes its "wave" pattern, and returns to the starting point. Understanding the period helps you predict how the graph behaves over different intervals.
The formula for the period of a sine function is given by: \[\text{Period} = \frac{2\pi}{|B|}\]where \( B \) is the coefficient of the variable \( x \) inside the function.
In the equation \( y = -2\sin\frac{2\pi x}{3} \), \( B \) is \( \frac{2\pi}{3} \). By applying the formula, the period is: \[\frac{2 \pi}{\frac{2 \pi}{3}} = 3\]
This tells us that the pattern of the sine function repeats every 3 units along the x-axis.

Graphing Sine Functions

Graphing sine functions involves understanding both amplitude and period, as they determine the size and spacing of the waves in the graph. With each sine function, these properties tell us how "tall" the waves are and how "long" each cycle lasts.
Here are some key points to help graph a sine function:

• Determine the Amplitude: The amplitude affects the wave's height, as discussed previously.
• Find the Period: This is the length over which the sine wave repeats, calculated using the period formula.
• Reflection: If there's a negative sign in front of the sine function, like in \( y = -2\sin\frac{2\pi x}{3} \), the graph reflects across the x-axis.

To graph \( y = -2 \sin \frac{2 \pi x}{3} \), you plot the x-values evenly over intervals of its period, 3, and adjust for the amplitude, 2. Don't forget the reflection aspect, causing the typical upward peaks to turn downward. By doing this, you can visualize how the function behaves over two full cycles, showcasing its repeating wave pattern visually.