Problem 27 - Calculus Single Variable

Question

Graph the function. \(f(t)=\sin (t-\pi / 6), 0 \leq t \leq 2 \pi\)

Step-by-Step Solution

Answer

The function is a sine wave shifted right by \\(\frac{\pi}{6}\\) with period \\(2\pi\\).

Step 1: Understand the basic function

The base function here is a sine function, \(y = \sin(t)\), which is a periodic wave with a period of \([0, 2\pi]\). This function has a maximum value of 1 and a minimum value of -1.

Step 2: Identify the transformation

The given function is \(f(t) = \sin(t - \frac{\pi}{6})\). This represents a horizontal shift (or phase shift) to the right by \(\frac{\pi}{6}\). This shift does not affect the amplitude or period of the wave.

Step 3: Set the new interval for graphing

Because of the horizontal shift, the start and end points of the interval within which the function will be graphed also shift. The original interval is \(0 \leq t \leq 2\pi\). Applying the shift, the effective interval becomes \(\frac{\pi}{6} \leq t \leq 2\pi + \frac{\pi}{6}\). However, for graphing from \(0\) to \(2\pi\), we consider the sine pattern starting from \(t = 0\) with the shift accounted as changes in starting points of the wave's features.

Step 4: Sketch the graph

When drawing the graph within \(0 \leq t \leq 2\pi\), note the shifted features. The main troughs, peaks, and zero crossings will occur at these modified points: \([\frac{\pi}{6}, \frac{7\pi}{6}]\), \([\frac{5\pi}{6}, \frac{11\pi}{6}]\), and so on. Always remember the sine wave pattern of crossing through zero at \(\pi, 2\pi, ...\).

Key Concepts

Sine Function Transformations

Transformations of the sine function involve altering its basic look without changing its fundamental sine wave nature. The basic sine function, noted as \( y = \sin(t) \), produces a smooth, wave-like graph. When transformations are applied, they change the graph's position, scale, or orientation, but they don't alter the core wave pattern.

Types of transformations include:

• Vertical shifts – Moving the graph up or down without affecting its shape.
• Horizontal shifts (phase shifts) – Sliding the wave left or right along the x-axis.
• Amplitude changes – Scaling the height of the wave, changing its maximum and minimum values.
• Period changes – Expanding or compressing the wave horizontally, affecting how quickly it repeats.

In our example, the transformation used is a horizontal shift, which we'll explore further. A vital part of understanding transformations is recognizing how each type affects the graph's unique features such as peaks, troughs, and zero points.

Periodic Functions

A function is considered periodic if it repeats its pattern over regular intervals. The sine function, \( y = \sin(t) \), is a classic example, completing one full cycle from \( 0 \) to \( 2\pi \). After \( 2\pi \), the function starts the same sine wave anew.

This inherent repetition makes sine functions extremely useful in modeling cyclic phenomena such as sound waves, tides, and electrical currents.

Key attributes of periodic functions:

• Period – The length of a complete cycle, often measured in radians (e.g., \( 2\pi \) for the sine function).
• Amplitude – The peak height from the centerline of the graph, often pointing to the maximum extent of change.
• Frequency – How often the wave repeats in a given interval.

Understanding periodicity helps in predicting future values and reactions, since these functions offer a regular, repeating backdrop against which changes are measured.

Phase Shift

The concept of phase shift is crucial for understanding the movement within periodic functions. A phase shift occurs when the entire graph of a function moves horizontally. In mathematical terms, it involves adding or subtracting a constant term from the independent variable \( t \).

For the function \( f(t) = \sin(t - \frac{\pi}{6}) \), the phase shift is \( \frac{\pi}{6} \) to the right. This means every feature of the sine wave—including its peaks, troughs, and zero crossings—will start \( \frac{\pi}{6} \) later than they would in the unshifted \( \sin(t) \).

Phase shifts do not alter the amplitude or period of the function. Therefore, the shape and size of the wave remain constant. Instead, the shift modifies where the wave starts in its cycle.

To find the new positions of important features within \( 0 \leq t \leq 2\pi \), you add the phase shift value to each critical point of the standard sine wave. For instance:

• Zero points, usually at \( 0, \pi, 2\pi \), become \( \frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6} \).
• Peaks, normally found at \( \frac{\pi}{2}, \frac{5\pi}{2} \), shift to \( \frac{\pi}{3}, \frac{11\pi}{6} \) within the graphing range.

Thus, phase shifts offer a strategic tool for modifying when and where wave features appear on the graph, allowing one to tailor the function's starting conditions to fit specific scenarios or data points.