Problem 130 - Precalculus

Question

Use a graphing calculator to graph \(Y_{1}=\sin x\) and \(Y_{2}=\sin (x+c),\) where a. \(c=\frac{\pi}{3},\) and explain the relationship between \(Y_{2}\) and \(Y_{1}\) b. \(c=-\frac{\pi}{3},\) and explain the relationship between \(Y_{2}\) and \(Y_{1}\)

Step-by-Step Solution

Answer

For \(c=\frac{\pi}{3}\), \(Y_2\) shifts left; for \(c=-\frac{\pi}{3}\), \(Y_2\) shifts right.

Step 1: Review the Functions

Identify the functions we need to analyze: the first function is \( Y_1 = \sin x \), which is the standard sine function. The second function is \( Y_2 = \sin (x + c) \), where \( c \) is a constant that will shift the sinusoidal waveform horizontally.

Step 2: Analyze for c=\(\frac{\pi}{3}\)

Plug \( c = \frac{\pi}{3} \) into the second function to get \( Y_2 = \sin(x + \frac{\pi}{3}) \). This represents a horizontal shift of the standard sine wave to the left by \( \frac{\pi}{3} \).

Step 3: Graph and Compare for c=\(\frac{\pi}{3}\)

Using the graphing calculator, graph \( Y_1 = \sin x \) and \( Y_2 = \sin(x + \frac{\pi}{3}) \). Observe that the graph of \( Y_2 \) is a leftward shift of \( \frac{\pi}{3} \) units compared to \( Y_1 \). This shift does not affect the amplitude or period of the graph, only its phase.

Step 4: Analyze for c=-\(\frac{\pi}{3}\)

Plug \( c = -\frac{\pi}{3} \) into the second function to get \( Y_2 = \sin(x - \frac{\pi}{3}) \). This indicates a horizontal shift of the standard sine wave to the right by \( \frac{\pi}{3} \).

Step 5: Graph and Compare for c=-\(\frac{\pi}{3}\)

Using the graphing calculator, plot \( Y_1 = \sin x \) and \( Y_2 = \sin(x - \frac{\pi}{3}) \). It can be observed that the graph of \( Y_2 \) is shifted to the right by \( \frac{\pi}{3} \) units compared to \( Y_1 \). Like the previous case, this horizontal shift affects only the phase.

Key Concepts

Phase Shift

In trigonometric functions, a **phase shift** is a horizontal shift along the x-axis. This shift affects the starting point of the function without changing its shape, amplitude, or period.When looking at sine functions, phase shifts occur in the form of adjustments to the angle inside the sine function. For example, consider the function:- \(Y_1 = \sin x\)- \(Y_2 = \sin(x + c)\)If \(c\) is positive, the function \(Y_2\) shifts left by \(c\) units compared to \(Y_1\). Conversely, if \(c\) is negative, the function shifts right by \(|c|\) units. In our specific exercise, we observed:- For \(c = \frac{\pi}{3}\), the graph shifts left by \(\frac{\pi}{3}\) units.- For \(c = -\frac{\pi}{3}\), the graph shifts right by \(\frac{\pi}{3}\) units.Understanding phase shifts is crucial in graphing as it doesn't alter the sine wave's amplitude or frequency, but only where it begins its cycle.

Sine Function

The **sine function** is a fundamental waveform in trigonometry noted for its regular oscillating pattern.This function is periodic with a standard equation of:- \(Y = \sin x\)**Key Characteristics of the Sine Function:**

• The graph of sine is a smooth wave which starts at the origin (0,0).
• The wave oscillates between 1 and -1, with an amplitude of 1.
• Its period, the length of one full cycle, is \(2\pi\).

When phase shifts are introduced, the sine function's basic properties remain deeply rooted:- The waveform's amplitude and period do not change even with phase shifts.- The starting point of its oscillation changes based on the phase shift parameter \(c\).The sine function is pivotal in understanding wave behavior across multiple disciplines such as physics and engineering.

Graphing Calculator

A **graphing calculator** is an essential tool for visualizing functions like sine waves. These calculators allow students to explore mathematical concepts interactively and intuitively.**Using a Graphing Calculator with Sine Functions:**- **Setup:** Enter the equations for the functions you wish to graph. For instance, input \(Y_1 = \sin x\) and \(Y_2 = \sin(x + c)\), where \(c\) can be any constant value.- **Graphing:** After inputting the functions, observe their graphical representation. Notice especially how the phase shift parameter affects the position of \(Y_2\) relative to \(Y_1\).

• For \(c = \frac{\pi}{3}\), the sine wave moves to the left.
• For \(c = -\frac{\pi}{3}\), observe the shift to the right.

- **Analysis:** Interpret the graphs to understand how shifts affect the function, and verify your observations with theoretical predictions.Graphing calculators are invaluable for students to confirm their textbook work in understanding the effects of phase shifts in trigonometric functions.