Problem 81 - 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. $$ y=\frac{1}{2} \tan \pi x $$

Step-by-Step Solution

Answer

The graph of the function \(y=\frac{1}{2} \tan \pi x \) displays a compressed tangent curve with a period of 1. There are asymptotes at the end of each period and the maximum and minimum values are \(y= \pm\frac{1}{2}\).

Step 1: Determine the Period

The function \(y=\frac{1}{2} \tan \pi x \) is a tangent function, which normally has a period of \(\pi\). The coefficient of x is \(\pi\), which affects the period of the function. Usually, the period of tangent function \(tan(bx)\) is given by \(\frac{\pi}{b}\). For this function, the period is \(\frac{\pi}{\pi}\) which equals 1.

Step 2: Identify the Scaling Factor

The coefficient of the tangent function is \( \frac{1}{2}\), thus the graph of the function will be vertically scaled by \(\frac{1}{2}\). This means the peak and valley of the function are at \(y= \pm\frac{1}{2}\). This results in a vertically compressed version of the general tangent function.

Step 3: Graph the Function

Using a graphing utility, graph the function \(y=\frac{1}{2} \tan \pi x \) over at least two periods which should be from x = 0 to x = 2 and from x = 2 to x = 4. Make sure your viewing window is properly scaled to see the function over these intervals. The graph should show typical tangent behavior, with asymptotes at the end of each period and crossing through the origin point of each period.

Key Concepts

Period of a Tangent Function

Understanding the period of a tangent function is crucial when trying to graph it. The period of a function refers to the distance along the x-axis for the function to complete one full cycle before it starts repeating. For the standard tangent function, \( \tan(x) \), the period is \( \pi \). That means every \( \pi \) units along the x-axis, the function repeats its pattern.

Now, when you see a function like \( y = \frac{1}{2} \tan(\pi x) \), the \( \pi \) inside the tangent function means that the period is modified. You can calculate the new period as \( \frac{\pi}{\pi} = 1 \), effectively shortening the cycle of the function to just 1 unit on the x-axis.

Vertical Scaling in Functions

Vertical scaling is a transformation that stretches or compresses a function's graph vertically. In the context of trigonometric functions, scaling affects the amplitude, or the 'height' of the peaks and valleys. The function \( y = \frac{1}{2} \tan(\pi x) \) has a scaling factor of \( \frac{1}{2} \), meaning that all the y-values are half of what they would be for a standard tangent function. Therefore, if the original \( \tan(x) \) has peaks at \( y = 1 \) and troughs at \( y = -1 \) for example, this scaled function will have its peaks at \( y = \frac{1}{2} \) and troughs at \( y = -\frac{1}{2} \) instead.

Graphing Utilities in Mathematics

Graphing utilities are tools that help visualize mathematical functions, making it easier to understand their behavior. They can be physical tools like graphing calculators or software programs and apps. When you need to plot a function like \( y = \frac{1}{2} \tan(\pi x) \) over at least two periods, a graphing utility allows you to input the equation and adjust the viewing window to display the necessary intervals on the x-axis and the appropriate range on the y-axis. Such utilities provide an accurate representation of functions, including features like asymptotes and intercepts, that might be complex to draw by hand.

Trigonometric Functions Graphing

Graphing trigonometric functions, like the tangent, involves understanding their unique characteristics – for instance, tangent functions have asymptotes where the function grows infinitely.

With our example \( y = \frac{1}{2} \tan(\pi x) \), plotting this function will show that it approaches infinity and negative infinity at regular intervals – the ends of each period – and crosses through the origin at the midpoint of each period. The graph will oscillate between peaks at \( y = \frac{1}{2} \) and troughs at \( y = -\frac{1}{2} \) within each one-unit period, providing a clear visual representation of how the function behaves over different intervals of the x-axis.