Problem 78 - Algebra and Trigonometry

Question

In Exercises 77–82, use a graphing utility to graph each function. Use a viewing rectangle that shows the graph for at least two periods. $$ y=\tan 4 x $$

Step-by-Step Solution

Answer

To graph \(y = \tan 4x\) in a way that shows at least two periods, select a viewing rectangle that spans from \(-\pi/4\) to \(\pi/4\) for \(x\) and from \(-10\) to \(10\) for \(y\). This choice ensures that the window captures two full periods of the function.

Step 1: Understanding the Graph of tangent function

The tangent function, \(y = \tan x\), has a period of \(\pi\). It contains vertical asymptotes at \(x = \pi/2 + k\pi\), where \(k\) is an integer and the function is undefined at these points.

Step 2: Adjusting for the Coefficient of \(x\)

The function given is \(y = \tan 4x\). The coefficient '4' in front of \(x\) will affect the period of the tangent function. The period of this function will be \(\pi/4\). The vertical asymptotes will occur at \(x = \pi/8 + k\pi/2\), where \(k\) is an integer, and the function will be undefined at these points.

Step 3: Selecting the Viewing Rectangle

To view at least two periods of the function, the viewing rectangle should span at least \(\pi/2\). Hence, an appropriate window for graphing this function might be \(-\pi/4\) to \(\pi/4\) for \(x\) and \(-10\) to \(10\) for \(y\). These values will ensure that the window captures at least two periods of the tangent function.

Step 4: Graphing the Function

Use the graphing utility to draw the function \(y = \tan 4x\) in the chosen viewing rectangle. Be careful to notice that the graph approaches vertical asymptotes every \(\pi/4\) and is undefined at these points.

Key Concepts

Tangent Function

The tangent function is a fundamental trigonometric function, often denoted by \( y = \tan x \). Unlike sine and cosine, the pattern of the tangent function repeats every \( \pi \), meaning it has a period of \( \pi \). This periodic nature is due to the tangent being undefined at certain points, specifically at \( x = \pi/2 + k\pi \), where \( k \) is an integer. These points are characterized by the tangent function approaching infinity as the graph nears them. The basic shape of the tangent graph involves rising steeply from negative infinity, crossing zero, and then climbing up towards positive infinity within each period. This unique structure makes tangent distinct and interesting among trigonometric functions.

Period of a Function

The period of a trigonometric function refers to the horizontal length over which the function’s shape repeats. For the standard tangent function \( y = \tan x \), the period is \( \pi \). This means that every \( \pi \) units along the x-axis, the pattern of the graph repeats itself. However, when the function changes, such as when coefficients are introduced to the argument of the tangent (e.g., \( y = \tan 4x \)), the period is affected. For \( y = \tan 4x \), the period becomes \( \pi/4 \). The period for a function like this can be determined by dividing the standard period by the coefficient of \( x \). This modification compresses the graph along the x-axis, allowing it to repeat more frequently within a given interval.

Vertical Asymptotes

Vertical asymptotes are lines that indicate where a function’s value tends to infinity. For the tangent function, these occur due to the undefined nature of the function at certain x-values. In its basic form, \( y = \tan x \) has vertical asymptotes at \( x = \pi/2 + k\pi \). When we consider functions like \( y = \tan 4x \), these asymptotes shift in position. They occur at \( x = \pi/8 + k\pi/2 \).

• As \( x \) approaches these values from the left or right, the graph of the tangent function will sharply rise or fall towards infinity.
• The positions of these asymptotes are crucially linked to the period of the tangent function, and knowing them helps in accurately sketching the graph and understanding the behavior of the function.

Trigonometric Graphing Utility

A trigonometric graphing utility is a tool like a graphing calculator or specific software that allows for the plotting and visualization of trigonometric functions such as the tangent. These utilities are essential for understanding complex functions where manual graphing might be tedious.

• They enable users to adjust the scale, which is important when exploring functions with non-standard periods like \( y = \tan 4x \).
• Using these utilities helps identify key characteristics like periods and vertical asymptotes, and also lets you explore different viewing rectangles to best capture the behavior of the function.
• For \( y = \tan 4x \), selecting a window like \(-\pi/4\) to \(\pi/4\) will allow you to see two full periods, which is important for fully visualizing the function's pattern.