Problem 21 - Algebra 2

Question

Sketch one cycle of each sine curve. Assume \(a>0 .\) Write an equation for each graph. amplitude \(1.5,\) period 3

Step-by-Step Solution

Answer

The equation for the sine curve with amplitude 1.5 and period 3 is \(y = 1.5 \sin(2\pi/3x)\). To sketch the graph, start from the origin and make it reach its maximum at \(3/4\), return to zero at 1.5, reach its minimum at 2.25, and return to the origin at 3 to complete one cycle.

Step 1: Identify the amplitude and the period

The amplitude given is 1.5, and the period is 3.

Step 2: Find the b value

Use the relationship between period and b, which is \(2\pi/b\), to find the b value. As the period given is 3, set \(2\pi/b = 3\), so \(b = 2\pi/3\).

Step 3: Write the equation of the sine function

Insert the values of the amplitude a and b into the general equation for the sine function. The equation of the sine function in this problem is \(y = 1.5 \sin(2\pi/3x)\).

Step 4: Sketch the graph of the sine function

The graph starts from the origin (0,0). It reaches the maximum value, which is the amplitude, at \(3/4\), then goes back to zero at 1.5, goes to the minimum at 2.25, and returns to 0 at 3, thus completing one cycle. Repeat this pattern to continue the graph.

Key Concepts

Understanding Amplitude

Amplitude is a fundamental concept in understanding trigonometric functions like the sine function. Imagine the wave-like pattern of a sine graph. The amplitude measures how tall or deep this wave goes from its central equilibrium position. Think of amplitude as the wave's height.

In mathematical terms, amplitude is the absolute value of the coefficient in front of the sine function. For example, in the function \(y = a \cdot \sin(bx)\), "a" represents the amplitude. If \(a = 1.5\), then the wave will rise 1.5 units above and fall 1.5 units below its midpoint or equilibrium line. This tells you how intense or strong the oscillation is.

To visualize this, let's break down a sine wave:

• The highest point the wave reaches is the positive amplitude: \(+1.5\)
• The lowest point the wave dips is the negative amplitude: \(-1.5\)
• The wave oscillates between these two points around the center line, which is usually the x-axis or \(y=0\)

Understanding amplitude helps in determining how the graph of the sine function looks and behaves.

Calculating the Period of a Function

The period of a function is another crucial aspect of understanding sine graphs. It tells you how long it takes for the wave to complete one full cycle. For the sine function, this means moving from one peak to the next corresponding peak or from one trough to the next corresponding trough.

The general formula to find the period of the sine function \(y = a \cdot \sin(bx)\) is \(\frac{2\pi}{b}\). Here, "b" affects how stretched or compressed the wave appears. If \(b\) changes, the period changes.

In our specific problem, the period is given as 3, which means it takes 3 units along the x-axis for the wave to repeat its pattern. To find \(b\), you rearrange and solve the formula:

• Set \(\frac{2\pi}{b} = 3\).
• Solve for \(b\): \(b = \frac{2\pi}{3}\).

This means that, in the equation, the value \(b = \frac{2\pi}{3}\) adjusts the wave to fit within a 3-unit interval on the x-axis for each complete cycle.

Working with Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and tangent. They're essential in modeling periodic phenomena, such as sound waves, light waves, and tides. Each trigonometric equation can tell us about specific characteristics of these waves, like their amplitude and period.

For example, consider the sine function constructed in the exercise: \(y = 1.5 \cdot \sin(\frac{2\pi}{3}x)\). Here’s how it works:

• The amplitude is 1.5, so the wave peaks at 1.5 and valleys at -1.5 relative to the x-axis.
• The period is derived from the formula \(\frac{2\pi}{b}\). Since \(b = \frac{2\pi}{3}\), the period simplifies to 3 units.
• Every value of x within these 3 units completes one wave cycle.

By solving and graphing trigonometric equations, you can visually display the oscillating pattern of real-world processes. Trigonometric equations also allow you to manipulate these patterns by changing amplitude or period, thereby modeling different scenarios.