Problem 77 - 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. $$ y=\tan \frac{x}{4} $$

Step-by-Step Solution

Answer

Using the graphing utility with x-axis marked with the intervals of \(π\), the graph is plotted by repeating the same shape that occurs between two vertical asymptotes over a period of \(4π\).

Step 1: Understand the function

The function given is a tan function \(tan(x/4)\) and we know that the normal period of tan function is \(\pi\). However, the presence of \(x/4\) in the function means that the period is now \(4\pi\). Find the critical points first, which are vertical asymptotes and zeros.

Step 2: Find the vertical asymptotes

The vertical asymptotes of a tangent function typically occur at \(±π/2,±3π/2,±5π/2, ...\) where the cosine function equals zero. However, in this case they will occur at \(\pm2π, \pm6π, \pm10π, ...\), which are \(4×\frac{π}{2}, 4×\frac{3π}{2}, 4×\frac{5π}{2}, ...\).

Step 3: Mark the zero points

The zeros of tan function usually occur at \(π×n\) where \(n\) is an integer. Here, the zero points will be when \(\frac{x}{4} = πn\), which means \(x = 4πn\), where \(n\) is an integer. So plot the zeros like \(0, 4π, 8π, -4π, -8π, ...\).

Step 4: Plotting the function

Using a graphing utility, plot the function using the critical points plotted in steps 2 and 3. Observe the shape in one period and draw the shape repeatedly for two periods or more.

Key Concepts

Graphing Utility

A graphing utility is an essential tool in visualizing mathematical functions, particularly when dealing with complex trigonometric functions like the tangent. Using a graphing utility allows students to effortlessly create a visual representation of the function across multiple periods. To graph the tangent function \(y=\tan \frac{x}{4}\), one would typically input the function into the utility, adjusting the viewing window to capture at least two periods. This helps in understanding the behavior of the function over a range of values, which is crucial when studying periodic functions.

It's important to familiarize oneself with the features of the graphing utility, including setting the appropriate domain, range, and scales for the axes. A good practice is to start with a standard viewing rectangle and adjust it as needed to ensure the critical points, such as zeros and vertical asymptotes, are clearly visible. Effective use of a graphing utility fosters a better conceptual grasp of the function's properties and its graphical characteristics.

Tangent Function Period

The period of a trigonometric function is the length of the smallest interval over which the function repeats its pattern. For a standard tangent function, \(y=\tan(x)\), the period is \(\pi\). However, when the independent variable is scaled, as in \(y=\tan(\frac{x}{4})\), the period is adjusted accordingly. In this case, the period becomes \(4\pi\), because dividing the variable by 4 stretches the function horizontally, causing it to complete one cycle over a larger interval.

Understanding the period is critical as it informs us about the function's repeating pattern and helps in identifying other properties, such as the location of zeros and asymptotes within each cycle. When graphing, depicting at least two periods can give insight into the consistency and predictability of the function, while also highlighting its symmetry and periodicity.

Vertical Asymptotes in Trigonometry

Vertical asymptotes in trigonometry represent the x-values at which functions like the tangent undergo infinite discontinuities. For the standard tangent function, asymptotes occur at \(\pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \pm\frac{5\pi}{2}, ...\), where the function is undefined because the cosine of these angles is zero. With the function \(y=\tan(\frac{x}{4})\), these asymptotes scale by the same factor as the period, appearing at \(\pm2\pi, \pm6\pi, \pm10\pi, ...\) instead.

Graphing these asymptotes is crucial as they delineate where the function's value 'jumps' from positive infinity to negative infinity, or vice versa. The tangent function will approach these lines without ever touching them, forming a characteristic 'break' in the graph. Students must be meticulous in plotting vertical asymptotes to gain a precise understanding of the function's behavior and ensure the graph's accuracy.