Problem 90 - Calculus for Biology and Medicine

Question

For each given pair of functions, use a graphing calculator to compare the functions. Describe what you see. \(y=\cos x\) and \(y=\cos 2 x\)

Step-by-Step Solution

Answer

\( y = \cos 2x \) completes two cycles per one cycle of \( y = \cos x \), indicating it has double the frequency.

Step 1: Understand the Functions

We have two functions to compare. The first is \( y = \cos x \), a basic cosine function. The second is \( y = \cos 2x \), which is a transformed cosine function where the angle \( x \) is multiplied by 2 within the function.

Step 2: Use a Graphing Tool

Plot both functions on a graphing calculator or software. You will see two wave-like graphs representing each function on a coordinate plane. On the x-axis, plot values over at least one full period, typically from \( 0 \) to \( 2\pi \), for clarity.

Step 3: Analyze the Graphs

Compare the two graphs. Note that the graph of \( y = \cos 2x \) completes one full cycle in a shorter interval compared to \( y = \cos x \). While \( \cos x \) completes one period over \( 2\pi \), \( \cos 2x \) completes a period over \( \pi \). Hence, \( y = \cos 2x \) has twice the frequency of \( y = \cos x \).

Step 4: Describe the Differences

Describe what you see: the function \( y = \cos x \) has its usual periodic nature with peaks and troughs occurring every \( 2\pi \). In contrast, \( y = \cos 2x \) oscillates more quickly, completing two full cycles within the interval from \( 0 \) to \( 2\pi \). This results in \( y = \cos 2x \) being more compressed in the horizontal (x) axis.

Key Concepts

Cosine function

The cosine function is one of the fundamental trigonometric functions used to model periodic phenomena. It is defined as the x-coordinate of a point on the unit circle—essentially, it describes how much shadow a circle's radius casts on the horizontal axis. The basic cosine function, represented as \( y = \cos x \), produces a wave-like pattern which is symmetrical and repeats every \( 2\pi \) radians.

Key characteristics of the cosine function include:

• Amplitude: The maximum height from the center line to the peak of the wave is 1.
• Period: The complete cycle of the function from start to finish, which for \( y = \cos x \) is \( 2\pi \).
• Range: The output values of the function range between -1 and 1.
• Symmetry: The cosine function is an even function, meaning \( \cos(-x) = \cos(x) \).

Periodicity

Periodicity refers to the property of functions to repeat values at regular intervals. In trigonometry, periodic functions like sine and cosine are commonplace. For the cosine function \( y = \cos x \), this means that the pattern of peaks and troughs repeats every \( 2\pi \) units along the x-axis. A complete cycle or period for \( y = \cos x \) is thus \( 2\pi \) radians, which is equivalent to 360 degrees.

When analyzing transformations of the cosine function, it's crucial to understand how the period changes. For example, in the function \( y = \cos 2x \), the coefficient 2 effectively compresses the period to \( \pi \) radians. This doubling of frequency results in the wave completing its cycle twice as fast within the same interval. Periodicity can be assessed by examining how the function repeats and identifying the length of one complete cycle.

• Normal Period: \( 2\pi \)
• Effect of Coefficient: The coefficient in front of x modifies the cycle's length as \( \frac{2\pi}{b} \), where \( b \) is the coefficient.

Graphing transformations

Graphing transformations involve altering a graph's shape or position without changing its basic type. For trigonometric functions, transformations can include translations, reflections, stretches, or compressions. When graphing \( y = \cos x \) and \( y = \cos 2x \), you'll notice that the latter appears more compressed along the x-axis.

Graphing transformations affecting the cosine function include:

• Horizontal Compression/Stretch: By changing the coefficient of x (e.g., \( y = \cos 2x \)), you adjust how many cycles fit into a given range.
• Vertical Shifts: Adjusting the graph's height up or down, although in our case functions primarily manipulate the x-values.
• Reflections: Flipping the graph over an axis, although this is more common with sine transformations.

It's important to plot these graphs carefully using a calculator or software when comparing transformations. This visualization aids in understanding how each alteration impacts periodicity and amplitude. By closely examining transformed graphs, you can better grasp the dynamic nature of trigonometric functions.