Problem 129 - Precalculus

Question

Use a graphing calculator to graph \(Y_{1}=\cos x\) and \(Y_{2}=\cos (x+c),\) 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

A positive shift (\(c = \frac{\pi}{3}\)) moves \(Y_{2}\) left, and a negative shift (\(c = -\frac{\pi}{3}\)) moves \(Y_{2}\) right relative to \(Y_{1}\).

Step 1: Initializing Graphing Calculator

Turn on the graphing calculator and enter the function for the first graph as \(Y_{1} = \cos x\). This serves as our reference cosine function.

Step 2a: Adjusting Function with Positive Phase Shift

Enter \(Y_{2} = \cos \left(x + \frac{\pi}{3}\right)\) into the graphing calculator. This represents the cosine function with a positive phase shift of \(\frac{\pi}{3}\). This will shift the graph of \(Y_{1}\) left by \(\frac{\pi}{3}\) units.

Step 3a: Observing the Positive Phase Shift

Graph \(Y_{1}\) and \(Y_{2}\) together. Observe how \(Y_{2}\) is a left shifted version of \(Y_{1}\). The phase shift is evident as all the peaks and valleys of \(Y_{2}\) occur \(\frac{\pi}{3}\) units before those of \(Y_{1}\).

Step 4a: Analyzing the Positive Shift Relationship

The function \(Y_{2}\) is periodic and has the same amplitude and wavelength as \(Y_{1}\), but it is shifted to the left by \(\frac{\pi}{3}\). This means \(Y_{2} = \cos(x + c)\) where \(c = \frac{\pi}{3}\) appears earlier than \(Y_{1}\).

Step 2b: Adjusting Function with Negative Phase Shift

Now adjust the second function to \(Y_{2} = \cos \left(x - \frac{\pi}{3}\right)\), which represents a negative phase shift of \(\frac{\pi}{3}\). This shifts the graph right by \(\frac{\pi}{3}\) units.

Step 3b: Observing the Negative Phase Shift

Graph \(Y_{1}\) and \(Y_{2}\) for the new \(Y_{2}\) function. Here, observe that \(Y_{2}\) as a right shifted version of \(Y_{1}\). Peaks and valleys occur \(\frac{\pi}{3}\) units after those of \(Y_{1}\).

Step 4b: Analyzing the Negative Shift Relationship

\(Y_{2}\) still has the same amplitude and period as \(Y_{1}\), but it is shifted right by \(\frac{\pi}{3}\). This shows that \(Y_{2} = \cos(x + c)\) where \(c = -\frac{\pi}{3}\) appears later than \(Y_{1}\).

Key Concepts

graphing calculator

A graphing calculator is an essential tool that simplifies the process of visualizing mathematical functions. It allows you to display multiple functions simultaneously, compare them, and detect shifts or variations easily. For trigonometric functions like cosine, the calculator can graph the periodic nature of the function which is helpful to understand transformations such as phase shifts.

When you enter a function, the graphing calculator plots the curve based on the equation you input. For example, by inputting \(Y_{1} = \cos x\), you get a visual representation of the standard cosine wave.

• Use the calculator's feature to add multiple functions: input \(Y_{2} = \cos(x + c)\) where \(c\) represents the phase shift.
• Experiment with different values for \(c\) to see how the graph changes.
• Observe how the graph's peaks and valleys shift horizontally, reflecting the effect of the phase shift.

Graphing calculators offer the advantage of immediate visual feedback on complex transformations, making them a vital educational tool for exploring the behavior of trigonometric functions.

cosine function

The cosine function \( \cos x \) is one of the primary trigonometric functions and is periodic in nature. It oscillates between -1 and 1, creating a smooth, wave-like pattern. This function repeats every \(2\pi\), which is its period.

The basic properties of the cosine wave include:

• Amplitude: The maximum height from the centerline is 1 and the minimum is -1.
• Period: As mentioned, the period is \(2\pi\) meaning it repeats every \(2\pi\) units.
• Symmetry: The cosine function is an even function, meaning it is symmetric around the y-axis, so that \(\cos(-x) = \cos(x)\).

These properties are not altered by transformations such as phase shifts. However, transformations do affect where and when these characteristic points appear on the graph, as with the concept of phase shifting, where the wave pattern is shifted left or right on the x-axis.

trigonometric graph transformations

Trigonometric graph transformations allow for the modification of graphs in various ways. One of the common transformations is the phase shift, which is a horizontal shift applied to the trigonometric function:

Phase shifts adjust the phase or "start point" of the wave. They are represented in the function as \(\cos(x + c)\) or \(\cos(x - c)\). Depending on the sign of \(c\), the graph shifts:

• Left Shift: \(c > 0\) results in a shift to the left. For instance, \(\cos(x + \frac{\pi}{3})\) moves the graph \(\frac{\pi}{3}\) units left, which means each point in the wave occurs earlier than in \(\cos x\).
• Right Shift: \(c < 0\) results in a shift to the right. For example, \(\cos(x - \frac{\pi}{3})\) shifts the graph \(\frac{\pi}{3}\) units to the right, making each characteristic point appear later.

These transformations maintain the overall shape, amplitude, and period of the function, ensuring that the wave's peaks and valleys remain consistent across adjustments. Understanding these shifts helps in predicting the behavior of trigonometric functions across different applications.