Problem 66 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the function. (Include two full periods.) Identify the amplitude and period of the graph. $$y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2$$

Step-by-Step Solution

Answer

The amplitude of the function is 3 and the period is 4.

Step 1: Identify the Amplitude

The amplitude of a trigonometric function is the absolute value of A. In the function \(y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2\), the amplitude is \(|3|\), which is 3.

Step 2: Identify the Period

The period of a trigonometric function is \(2\pi/B\) . Here, B is \( \frac{\pi}{2} \), so the period is \( 2\pi/(\pi/2) = 4\).

Step 3: Graph the function

Using a graphing utility, draw the cosine function with the specified amplitude and period over two full periods. Mark clearly the maxima and minima value of the function on y-axis which is equal to \(D + |A|\) and \(D - |A|\) respectively, and period/4, period/2, 3*period/4 and period on x-axis because cosine function has its maximum at 0, its minimum at period/2 and again it is at maximum at period. Here maxima and minima are \(|-2 + 3| = 1\) and \(|-2 - 3| = -5\), and the period is 4 so we mark 1, 2, 3 and 4 on x-axis. Draw the graph of function properly with above specifications.

Key Concepts

Amplitude

In trigonometric functions like sine and cosine, the amplitude represents how much the graph of the function stretches or shrinks vertically. It tells you the height from the center line of the graph to the peak or trough. Amplitude is crucial in understanding the overall shape and dynamics of these graphs.

For the cosine function given by the exercise, \(y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2\), determining the amplitude involves identifying the absolute value of the coefficient before the cosine function. In this case, the coefficient is \(3\); therefore, the amplitude is \(3\).

• This means that each peak of the waveform will reach 3 units above the center horizontal line, and every trough will go 3 units below it.
• The center line is shifted down by \(-2\), which affects where these peaks and valleys actually occur on the graph.

Period

The period of a trigonometric function defines how long it takes for the function to complete one full cycle before repeating itself. In our function, the period indicates the horizontal length of one full wave on the graph.

To find the period of the function \(y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2\), use the formula \(2\pi/B\), where \(B\) is the coefficient of \(x\) in the angle. Here, \(B= \frac{\pi}{2}\). Calculating this gives:

• \(2\pi/ \frac{\pi}{2} = 4\)

Thus, the period is 4.

• This means the wave repeats every 4 units along the x-axis.
• It's essential as it allows us to understand spacing on the x-axis, especially when plotting or interpreting the graph.

Graphing Utility

A graphing utility, whether a calculator, computer program, or app, is a powerful tool used to visualize mathematical functions. It becomes particularly handy in plotting trigonometric graphs, where precision in amplitude and period is crucial.

When using a graphing utility to plot \(y=3 \cos \left(\frac{\pi x}{2}+\frac{\pi}{2}\right)-2\), ensure:

• You adjust settings to showcase at least two full periods for a complete understanding of the cycle pattern. With a period of 4, this would mean setting the x-axis scale from, say, 0 to 8.
• Identify and mark critical points like maxima and minima. Maxima occur at \(1\), and minima occur at \(-5\). These derive from the amplitude shifted by the vertical translation \(-2\).
• Also, ensure that the graph's resolution (the fineness of its grid) is high enough to clearly depict its peaks and troughs.

Using a graphing utility can simplify and speed up the process of understanding these graphs by providing visual confirmation of mathematical computations.