Problem 55 - 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.) $$y=\frac{2}{3} \cos \left(\frac{x}{2}-\frac{\pi}{4}\right)$$

Step-by-Step Solution

Answer

The graph of the function is a vertically shrunk and horizontally stretched cosine wave that is shifted \(\frac{\pi}{4}\) units to the right. The peak values are at \(\frac{2}{3}\) and \(-\frac{2}{3}\). The period is \(4 \pi\) instead of \(2 \pi\).

Step 1: Identify Function Parameters

The given function is \(y=\frac{2}{3} \cos\left(\frac{x}{2}-\frac{\pi}{4}\right)\). The number multiplying the cosine function, which is \(\frac{2}{3}\), is the amplitude. The term inside the cosine function affects the horizontal shift and the frequency. The number multiplying \(x\), which is \(\frac{1}{2}\), changes the frequency of the function, and the constant term \(-\frac{\pi}{4}\) is a phase shift.

Step 2: Draw the Basic Cosine Graph

Start by drawing the basic cosine function \(y=\cos(x)\), which is a wave that peaks at 1 and goes down to -1, completing a cycle every \(2\pi\) units.

Step 3: Apply the Frequency and Amplitude

Scale the x-axis by the frequency which is \(2\) (the reciprocal of \(\frac{1}{2}\)), meaning, a cycle will now be completed every \(4 \pi\) units instead of \(2 \pi\). This stretches the cosine function horizontally. Then, scale the y-axis by the amplitude \( \frac{2}{3} \), meaning the maximum and minimum values of the function would be \(\frac{2}{3}\) and \(-\frac{2}{3}\) respectively. This shrinks the cosine function vertically.

Step 4: Apply the Phase Shift

Shift the graph to the right by \(\frac{\pi}{4}\) units to represent the phase shift. The peak of the function which initially was at \(x=0\) will now be shifted to \(x=\frac{\pi}{4}\).

Step 5: Verify With a Graphing Utility

Use a graphing calculator or any other graphing software or online tool to check the accuracy of the graph. Input the function \(y=\frac{2}{3} \cos\left(\frac{x}{2}-\frac{\pi}{4}\right)\) and check whether the graph on the screen matches the sketch.

Key Concepts

Amplitude of Trigonometric Functions

The amplitude of a trigonometric function describes how tall or short its waves are. It represents the maximum value that the function can reach above or below its midline, which is usually the x-axis. For a standard cosine or sine function, like y = cos(x) or y = sin(x), the amplitude is 1. However, when a coefficient is multiplied by the trig function as in y = A cos(x) or y = A sin(x), the amplitude becomes |A|.

In our exercise, the function y = \(\frac{2}{3}\) cos(x) has an amplitude of \(\frac{2}{3}\). This means the graph will reach a maximum height of \(\frac{2}{3}\) above the midline and a maximum depth of \(\frac{2}{3}\) below it. When graphing, you adjust the peaks and troughs to this new amplitude, creating a 'flatter' wave compared to the standard cosine function.

Frequency of Trigonometric Functions

Frequency in the context of trigonometric functions refers to the number of cycles the function completes in a given interval, typically 2\pi for the sine and cosine functions. The fundamental frequency occurs when there is no coefficient altering the x variable, such as in y = cos(x), where the function completes one cycle every 2\pi units.

When there is a coefficient, like y = cos(Bx), you can find the new period by calculating \(\frac{2\pi}{|B|}\), and the frequency is inversely related to the period. For our function, y = \(\frac{2}{3}\) cos(\(\frac{x}{2}\)), the period becomes \(4\pi\), which means the frequency is halved compared to the standard cosine function. The function is 'stretched' horizontally, leading to fewer cycles within the same interval.

Phase Shift in Trigonometry

Phase shift is a horizontal displacement of a trigonometric graph along the x-axis. In a function like y = cos(x - C) or y = sin(x - C), the value of C tells us how much the entire graph is shifted to the right (if C is positive) or to the left (if C is negative).

For our function, y = \(\frac{2}{3}\) cos(\(\frac{x}{2} - \frac{\pi}{4}\)), the phase shift is \(\frac{\pi}{4}\) units to the right because the term inside the cosine function subtracts a positive \(\frac{\pi}{4}\). This means every point on the graph of y = \(\frac{2}{3}\) cos(\(\frac{x}{2}\)) without the phase shift is moved \(\frac{\pi}{4}\) units to the right to get the graph of our given function.

Graphing Cosine Functions

Graphing cosine functions begins with understanding the standard cosine curve, which starts at its maximum value at x = 0, descends to 0 at \(\frac{\pi}{2}\), reaches its minimum at \(\pi\), ascends back to 0 at \(\frac{3\pi}{2}\), and completes the cycle at 2\pi. Variations in amplitude, frequency, and phase shift alter this standard graph.

For the given function, y = \(\frac{2}{3}\) cos(\(\frac{x}{2} - \frac{\pi}{4}\)), the graphing steps are adjusted as follows:

• Scale the amplitude to \(\frac{2}{3}\).
• Adjust the period to 4\pi due to the \(\frac{1}{2}\) frequency factor.
• Shift the graph to the right by \(\frac{\pi}{4}\).

To verify the accuracy of these adjustments, comparing the sketch with a graphing utility helps ensure the shifts were applied correctly.