Problem 169 - Precalculus

Question

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes. $$ f(x)=\frac{1}{4} \sin x $$

Step-by-Step Solution

Answer

Amplitude: \( \frac{1}{4} \), Period: \( 2\pi \), Midline: \( y = 0 \), Asymptotes: None.

Step 1: Identify the function type and basic features

The given function is \( f(x) = \frac{1}{4} \sin x \), which is a sine function. We will identify the amplitude, period, and midline from the function's form. The function represents a sine wave with modifications to the amplitude.

Step 2: Determine the amplitude

The amplitude of a sine function \( f(x) = a \sin x \) is given by \( |a| \). Here, the amplitude is \( \left| \frac{1}{4} \right| = \frac{1}{4} \). This means the wave's maximum displacement from the midline is 0.25.

Step 3: Calculate the period

The period of a standard sine function \( \sin x \) is \( 2\pi \). Since there is no horizontal scaling factor in \( f(x) = \frac{1}{4} \sin x \), the period remains \( 2\pi \). The function completes one full cycle in an interval width of \( 2\pi \).

Step 4: Determine the midline equation

The midline of a sine function \( f(x) = a \sin x + d \) is \( y = d \). The function \( f(x) = \frac{1}{4} \sin x \) has no vertical shift \((d = 0)\), so the midline equation is \( y = 0 \).

Step 5: Identify asymptotes for a sine function

Sine functions like \( \frac{1}{4} \sin x \) do not have vertical asymptotes because the sine function is continuous and periodic without undefined values.

Step 6: Graph the function over two periods

To graph \( f(x) = \frac{1}{4} \sin x \) over two periods (\( 0 \leq x \leq 4\pi \)), plot points at key positions like 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \). The graph oscillates between \( -\frac{1}{4} \) and \( \frac{1}{4} \), with a midline at \( y = 0 \) and repeating every \( 2\pi \).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that describes a smooth, periodic oscillation. When graphing the sine function, you can visualize waves, which is why it is often associated with cycles in nature, like sound or water waves. For a function in the form of \( f(x) = a \sin(bx + c) + d \), the sine function holds key characteristics that define its shape: amplitude, period, phase shift, and vertical shift.

• **Amplitude** determines how tall or short the waves appear.
• **Period** indicates how long it takes to complete a single cycle of the wave.
• **Midline** or vertical shift changes the center line of the wave up or down.

These properties help in graphing and understanding how the sine wave behaves over a certain interval.

Amplitude

Amplitude is a measure of how far the peaks and troughs of the wave are from the midline. In mathematical terms, it is expressed as the absolute value of the coefficient \( a \) in the sine function \( f(x) = a \sin(x) \).
For example, in the function \( f(x) = \frac{1}{4} \sin(x) \), the amplitude is \( |\frac{1}{4}| = 0.25 \). This shows that:

• The wave oscillates between \( -0.25 \) and \( 0.25 \).
• Amplitude affects the maximum and minimum points of the graph, providing a clear boundary for where the wave will extend.

Understanding amplitude is crucial when measuring the intensity or energy of waves, such as in physics and engineering applications.

Period

The period of a sine function defines the length of one complete wave cycle. It is determined by the coefficient \( b \) in \( f(x) = \sin(bx) \), calculated using the formula \( \frac{2\pi}{b} \).
For the function \( f(x) = \frac{1}{4} \sin(x) \), which has no coefficient other than 1 for \( x \), the period remains the standard \( 2\pi \). This implies that:

• The wave repeats itself every \( 2\pi \) units along the x-axis.
• You would see identically shaped cycles every \( 2\pi \) distance, such as from 0 to \( 2\pi \) and again from \( 2\pi \) to \( 4\pi \).

A precise understanding of the period is essential for applications requiring timing and synchronization of waves, like in radio frequencies or signal processing.

Midline

The midline of a sine function represents the horizontal line that runs through the middle of the waveform, around which the wave oscillates. This is given by \( y = d \) in the function \( f(x) = a \sin(bx) + d \). For the function \( f(x) = \frac{1}{4} \sin(x) \), the midline is \( y = 0 \) because there is no vertical shift added to the function.

• The midline serves as a baseline for measuring amplitude.
• In applications, it's often a reference level, like the average level in alternating electrical currents.

Knowing the midline helps in predicting the average value of a wave, which is practical in various scientific and engineering fields.

Graphing

Graphing sine functions entails plotting their oscillatory behavior over a specified range or period. For \( f(x) = \frac{1}{4} \sin(x) \), we want to plot it over two periods, or from 0 to \( 4\pi \).
Key points to plot:

• Start at \( (0, 0) \), indicating the wave starts on the midline.
• The maximum at \( \frac{\pi}{2} \) gives \( (\frac{\pi}{2}, \frac{1}{4}) \).
• Return to midline at \( \pi \), giving \( (\pi, 0) \).
• The minimum at \( \frac{3\pi}{2} \) is \( (\frac{3\pi}{2}, -\frac{1}{4}) \).
• Finally, at \( 2\pi \), back to \( (2\pi, 0) \).

Repeat this pattern for the second period from \( 2\pi \) to \( 4\pi \). By following these steps, you can accurately illustrate the sine wave's periodic nature, crucial for visual learners and practical applications in signal transmission or audio engineering.