Question

Make a complete graph of each function. $$y=\tan \left(x-\frac{\pi}{2}\right)$$

Short answer

The graph of \(y = \tan\left(x - \frac{\pi}{2}\right)\) has vertical asymptotes at \(x = k\pi\) and passes through the points \(x = -\frac{\pi}{2}\), \(x = 0\), and \(x = \frac{\pi}{2}\) after computing the corresponding \(y\) values.

Step-by-step solution

Understanding the Function

Identify the basic function involved. The function given is the tangent function, which has the form \(y = \tan(x)\). This function has a period of \(\pi\) and vertical asymptotes at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is any integer. The graph of the tangent function repeats every \(\pi\) units and is undefined at its vertical asymptotes.

Apply the Horizontal Shift

Recognize that \(y = \tan\left(x - \frac{\pi}{2}\right)\) is a horizontal shift of the basic tangent graph. The entire graph is shifted to the right by \(\frac{\pi}{2}\) units. This means that the vertical asymptotes of the graph will also shift right by \(\frac{\pi}{2}\) units, and the new vertical asymptotes will be at \(x = k\pi\).

Identifying the New Asymptotes

The new vertical asymptotes occur where the argument of the tangent function \(x - \frac{\pi}{2}\) is equal to \(\frac{\pi}{2} + k\pi\). Thus, the asymptotes are located at \(x = k\pi\), where \(k\) is any integer.

Determine the Period and Points

The period of the function remains the same, \(\pi\). Choose points to plot on the graph that are within one period of the function and do not lie on an asymptote. For example, select points such as \(x = -\frac{\pi}{2}\), \(x = 0\), and \(x = \frac{\pi}{2}\). Compute the corresponding \(y\) values for each selected \(x\) to identify key points through which the function passes.

Sketching the Graph

Use the plotted points and asymptotes to sketch the function. The graph will resemble a series of curves that pass through the points and approach the vertical asymptotes. Be sure to represent the repeating nature of the tangent function by drawing similar curves for each period.

Key concepts

Tangent Function Transformations

Graphing the tangent function often involves applying transformations to its basic form, which is expressed as \(y = \tan(x)\). Transformations can include shifts, stretches, compressions, and reflections, all of which change the appearance of the graph without altering its fundamental characteristics.

When we examine the given function \(y = \tan\left(x - \frac{\pi}{2}\right)\), we're dealing with a horizontal shift. Specifically, the graph of the tangent function is shifted to the right by \(\frac{\pi}{2}\). Such a shift can be visualized by imagining that every point on the basic tangent graph slides along the x-axis by \(\frac{\pi}{2}\) units. This horizontal transformation is crucial in determining the new position of the vertical asymptotes and the shape of the function's graph.

To grasp this concept, imagine that the original tangent function is a template. Each transformation we apply moves or reshapes this template while maintaining its intrinsic qualities. Understanding transformations empowers you to graph complex trigonometric functions with confidence and precision.

Period of Tangent Function

The period of a function is the length of the interval over which the function repeats its pattern. For the basic tangent function, \(y = \tan(x)\), this period is \(\pi\) radians. No matter how we transform the tangent function, its period remains unchanged unless we apply a horizontal stretch or compression.

In our exercise, the transformed function is not affected by a change in period because it only involves a horizontal shift. This means the pattern of the function—how it rises, falls, and approaches its asymptotes—repeats every \(\pi\) radians, just like the basic tangent function. It's vital to remember this characteristic when plotting points and sketching the curves of the tangent graph. Identifying correctly spaced key points within one period creates an accurate representation of the function's behavior and ensures a precise graph.

Once the period is understood and the key points are plotted, the repeating nature of the graph can be copied from one interval to the next, making the task of graphing much easier. You can then focus on other details such as symmetry and asymptotes without worrying about the fundamental shape and spacing of the function's curves.

Vertical Asymptotes in Trigonometry

Vertical asymptotes are lines where a function's value grows infinitely large or decreases infinitely large as it approaches the line, but never crosses or touches it. In the realm of trigonometry, vertical asymptotes occur in the tangent function because it becomes undefined at certain points.

In the basic form of the tangent function, the vertical asymptotes are located at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is an integer. These asymptotes represent the edges of the regions where the tangent function is defined. Whenever you are given a tangent function that has been transformed, like the one in our exercise \(y = \tan\left(x - \frac{\pi}{2}\right)\), identifying the new vertical asymptotes is key to understanding the function's graph.

For our transformed function, the asymptotes shift right by \(\frac{\pi}{2}\), and consequently, they are positioned at \(x = k\pi\). Once you’ve determined the locations of these asymptotes, drawing the graph becomes much easier. You must ensure that the function approaches these lines without touching or crossing them, as they act as barriers that delineate the behavior of the function. Recognizing and accurately illustrating vertical asymptotes in trigonometric functions is a fundamental skill that contributes greatly to your success in graphing these functions.