Problem 48 - Algebra 2

Question

Sketch one cycle of each sine curve. Assume that \(a>0 .\) Then write an equation for each graph. amplitude \(1,\) period \(\frac{\pi}{3}\)

Step-by-Step Solution

Answer

The sketch should be a sine wave starting at 0, reaching a height of 1, then continuing below to -1, and the whole cycle repeats within a period of \(\frac{\pi}{3}\). The equation for the given parameters is \(y = \sin(\frac{6}{\pi}x)\).

Step 1: Understand and Identify Key Parameters

First, identify the key parameters: amplitude and period. In this case, the amplitude is 1, and the period is \(\frac{\pi}{3}\). It implies that every \(\frac{\pi}{3}\) radians, the graph completes one cycle, and the maximum height (absolute value) that it reaches is 1.

Step 2: Plotting the Basic Sine Curve

Next, start by plotting a basic sine wave. Normally, the sine function starts from 0 at \(x=0\), rises to 1 at \(x=\frac{\pi}{2}\), goes back to 0 at \(x=\pi\), drops to -1 at \(x= \frac{3\pi}{2}\), and finally returns to 0 at \(x=2\pi\).

Step 3: Adjusting the Sine Curve

The task requires a cycle within the period of \(\frac{\pi}{3}\) instead of \(2\pi\). So, shrink the x-axis by a factor of \(\frac{\frac{\pi}{3}}{2\pi} = \frac{6}{\pi}\), to make the cycle complete at \(\frac{\pi}{3}\). The curve from \(x=0\) to \(x=\frac{\pi}{3}\) gives the required sine curve.

Step 4: Writing the Equation

Finally, write an equation for the graph. The general formula for a sine function is \(y = a \sin(bx)\) where `a` denotes amplitude and `b` is related with the period in the way that the period is \(2\pi/b\). As the sine curve now completes a cycle every \(\frac{\pi}{3}\), the equation becomes \(y = \sin(\frac{6}{\pi}x)\).

Key Concepts

Amplitude

The amplitude of a sine wave represents how tall or short the wave is in terms of its height from the centerline. Think of it as the measure of how "high" or "low" the wave gets.
In mathematical terms, amplitude is defined as the maximum absolute value of the wave from its midpoint. For a standard sine wave, the amplitude can be altered using a multiplier.
In the equation of a sine wave, \(y = a \sin(bx)\), the value of \(a\) determines the amplitude.
Here are some key points about amplitude:

• If \(a = 1\), the wave reaches a height of 1 and a depth of -1.
• If \(a > 1\), the wave gets taller. For example, \(a = 2\) means the wave reaches heights of 2 and depths of -2.
• If \(0 < a < 1\), the wave becomes shorter.
• Amplitude is always positive because it refers to magnitude, not direction.

In our exercise, the amplitude is 1, which means the sine wave will oscillate between 1 and -1.

Period

The period of a sine wave is the length of one complete cycle of the wave. It tells you how long it takes for the sine function to repeat its values.
More simply, the period is the distance required along the x-axis for a sine pattern to start over.
In the sine function equation \(y = a \sin(bx)\), the period of the sine wave is given by \(\frac{2\pi}{b}\).
Some fundamental points to remember about periods include:

• The standard period of a sine wave is \(2\pi\). This corresponds to the usual sine wave that completes a cycle as \(x\) progresses 0 to \(2\pi\).
• If \(b > 1\), the period decreases, causing the wave to complete faster cycles. Essentially, the x-axis becomes more "squished."
• If \(0 < b < 1\), the period increases, meaning the wave takes longer to complete a cycle or "stretches" along the x-axis.

In the given exercise, the period is \(\frac{\pi}{3}\), indicating the sine wave goes through one full cycle in that distance along the x-axis.

Sine Function Equation

The sine function equation is a mathematical representation of a sine wave, typically written in the form \(y = a \sin(bx)\). This equation provides a way to model periodic phenomena.
Each part of the equation contributes to shaping the sine wave:

• The coefficient \(a\) changes the amplitude, or height, of the wave.
• The variable \(b\) influences the wave’s period and determines how compressed or stretched the wave is along the x-axis.

Understanding how these parameters work lets you accurately graph sine waves for different scenarios. In our specific problem, with an amplitude of 1 and a period of \(\frac{\pi}{3}\), we identify \(a = 1\) and find \(b\) such that the period is \(\frac{2\pi}{b} = \frac{\pi}{3}\). Solving for \(b\), we have \(b = \frac{6}{\pi}\). Thus, the equation of the sine wave becomes \(y = \sin\left(\frac{6}{\pi}x\right)\).
This compact equation encapsulates the wave's behavior, guiding you in sketching and understanding its complete shape.