Problem 96 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

\(y = \tan (2x)\) has a shorter period and oscillates more frequently than \(y = \tan x\).

Step 1: Understanding the Functions

The given functions are \(y = \tan x\) and \(y = \tan (2x)\). Both functions are variations of the tangent function, but \(y = \tan (2x)\) has a modified argument.

Step 2: Graphing on the Calculator

Use a graphing calculator to input and plot both functions. Set your calculator to radian mode, as trigonometric functions are typically considered in radians.

Step 3: Observing the Basic Tangent Function

In the graph of \(y = \tan x\), observe that the function has vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. The function repeats every \(\pi\) units.

Step 4: Observing the Transformed Tangent Function

For the graph of \(y = \tan (2x)\), notice that the period of the function is reduced. The vertical asymptotes now occur at \(x = \frac{\pi}{4} + k\frac{\pi}{2}\), indicating a period of \(\frac{\pi}{2}\), due to the factor of 2 inside the tangent function.

Step 5: Comparing the Functions

The graph of \(y = \tan (2x)\) compresses horizontally compared to \(y = \tan x\), resulting in more frequent oscillations due to its shorter period. Both functions have the same amplitude but differ in the frequency of their cycles.

Key Concepts

Using a Graphing Calculator

When working with trigonometric functions, a graphing calculator is a helpful tool to visualize how these functions behave. By plotting different functions on a graphing calculator, you can observe changes and compare them in real-time.
If you're new to using graphing calculators, here are some tips to get started:

• Ensure the calculator is set to radian mode, as this is the standard for trigonometric functions.
• Input the functions correctly, paying attention to any transformations or coefficients.
• Use the zoom feature to get a better look at specific sections of the graph, especially where there are rapid changes or key features like asymptotes.

Graphing calculators make it easier to identify features like vertical asymptotes and the periodic nature of trigonometric functions.

Understanding the Tangent Function

The tangent function, expressed as \(y = \tan x\), is a fundamental trigonometric function. It has some unique properties that differ from sine and cosine.
Unlike sine and cosine functions, which have a range between -1 and 1, the tangent function can take any real value. This happens because tangent represents the ratio of sine to cosine:

• \(\tan x = \frac{\sin x}{\cos x}\)
• It creates vertical asymptotes at the points where \(\cos x = 0\), leading to undefined values.

These asymptotes appear periodically, contributing to the function's repeating pattern.
The basic period of the tangent function is \(\pi\) (approximately 3.14), meaning it repeats its shape after a distance of \(\pi\) along the x-axis. This periodic property makes it useful in various mathematical and practical applications.

Exploring the Period of a Function

The period of a trigonometric function is the distance over which the function repeats itself. For the basic tangent function, \(y = \tan x\), the period is \(\pi\).
When transformations are applied, like in the function \(y = \tan(2x)\), the period changes.

• For \(y = \tan(2x)\), the function's period becomes \(\frac{\pi}{2}\).
• This occurs because the "2" inside the function scales the input, causing the function to oscillate more rapidly.

The concept of period is crucial in understanding how different transformations affect the behavior of trigonometric functions.
Recognizing changes in period helps in graphing these functions accurately and predicting their behavior over different intervals.

Identifying Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. For the tangent function, these occur because the function is undefined where the cosine of the angle equals zero.
This is evident in the graph of \(y = \tan x\), which has vertical asymptotes at:

• \(x = \frac{\pi}{2} + k\pi\)

In \(y = \tan(2x)\), the vertical asymptotes occur more frequently due to the compressed period:

• \(x = \frac{\pi}{4} + k\frac{\pi}{2}\)

These asymptotes divide the graph into segments where the function displays characteristic curves. Understanding the location of vertical asymptotes is essential in graphing tangent functions and predicting where function values become extremely large or small.