Problem 48 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function. Use a graphing utility to verify your sketch. (Include two full periods.) $$y=\sin \frac{x}{4}$$

Step-by-Step Solution

Answer

The function y=sin(x/4) is a horizontally stretched version of the basic sine function y=sin(x), with a period of \(8\pi\) rather than \(2\pi\). After sketching, confirm the graph using a graphing utility.

Step 1: Understanding the Basic Sine Function

The basic sine function, y = sin(x), has a period of \(2\pi\). This means the graph repeats itself once every \(2\pi\) units.

Step 2: Understanding the Given Function

In the function y=sin(x/4), the 'x' input is divided by 4 before the sine function is applied. This is equivalent to stretching the graph of the basic sine function horizontally by a factor of 4. So the period is now \(2\pi \times 4 = 8\pi\).

Step 3: Sketching the graph

At x=0, sin(0)=0, so the graph starts at the origin (0,0). The graph will reach its first peak at \(x = 2\pi\), where y=1. Then, it returns to 0 at \(x = 4\pi\), goes to its lowest point at \(x = 6\pi\) where y=-1, and finally, completes its first full cycle at \(x = 8\pi\) where y=0 again. To include two full periods, the graph should continue this pattern until \(x = 16\pi\).

Step 4: Verifying the graph with a graphing utility

After sketching the graph, verify it using a graphing utility. Input the function y=sin(x/4) into the utility and compare the output with the sketched graph. The peaks, troughs and zero crossings should all match the sketched graph.

Key Concepts

Understanding the Sine Function

The sine function, represented as \(y = \sin(x)\), is one of the fundamental trigonometric functions that describes a smooth, wave-like motion. It oscillates between -1 and 1, creating a familiar wave pattern. This wave pattern repeats every \(2\pi\) units, which is referred to as its period.

• Zero Points: Occur where the sine function crosses the x-axis, i.e., at \(x = 0, \pi, 2\pi,...\)
• Maximum Points: Occurs at \(x = \frac{\pi}{2}, \frac{5\pi}{2},...\), where the waveform reaches a value of 1.
• Minimum Points: Occurs at \(x = \frac{3\pi}{2}, \frac{7\pi}{2},...\), where the waveform dips to -1.

When transforming sine functions, such as \(y = \sin\left(\frac{x}{4}\right)\), the period can change due to changes in the frequency of oscillation. Such transformations allow for a variety of wave patterns to be modeled, each displaying unique characteristics and periodicity.

The Role of Graphing Utilities

Graphing utilities are powerful tools that help visualize mathematical functions quickly and accurately. For complex and transformed functions like \(y = \sin\left(\frac{x}{4}\right)\), it is beneficial to use a graphing calculator or graphing software to check your work.

• Input the Function: Enter the function into the utility to generate a graph.
• Compare Characteristics: Verify key features such as peaks and zero-crossings against your hand-drawn sketch.
• Adjust Parameters: Experimentation with adjustments can provide an intuitive understanding of how transformations like stretching or compressing affect the function.

Using graphing utilities, you can also examine features beyond the standard range or spot errors in your graph. They serve as an excellent way to visually analyze and confirm the behavior of the function across multiple periods.

What Makes a Function Periodic?

A periodic function repeats its values at regular intervals, identified by its period. The sine function is a classic example, repeating every \(2\pi\) as it returns to its starting point after one full cycle.

• Basic Periodic Pattern: The repeated sequence includes rising to a peak, falling to a trough, and returning to its original state.
• Period Calculation: For \(y = \sin\left(\frac{x}{4}\right)\), the period changes, calculated by multiplying the standard period \(2\pi\) by the reciprocal of the coefficient of \(x\), yielding \(8\pi\).
• Real-World Applications: Periodic functions model natural phenomena such as sound waves, tides, and seasonal patterns.

By understanding periodicity, we can predict and visualize behaviors over time, essential for modeling cyclic events in both natural and technical environments.