Question

WRITING Explain why there is more than one tangent function whose graph passes through the origin and has asymptotes at \(x=-\pi\) and \(x=\pi\).

Short answer

The graph of every function of the form \(y = \tan(x + k\pi)\), where \(k\) is an integer, passes through the origin and has asymptotes at \(x=-\pi\) and \(x=\pi\). This is due to the periodic nature of the tangent function which repeats itself every \(\pi\) units.

Step-by-step solution

Understanding the behavior of the tangent function

Firstly, we need to understand that the graph of the tangent function has asymptotes at values of \(x\) for which the cosine function equals zero, i.e., at \(x=\frac{(2n-1)\pi}{2}\) for every integer \(n\). Between every two such values of \(x\), the tangent function increases from \(-\infty\) to \(\infty\). If you were to draw out this pattern on a graph, you would see that the graph of the tangent function crosses through the origin and repeats itself every \(\pi\) units, having an asymptote at \(x=\pm\pi, \pm 2\pi, \ldots\).

The periodicity of the tangent function

Due to the inherent nature of the tangent function to repeat itself after every \(\pi\) units, there can exist multiple tangent functions that pass through the origin and have asymptotes at \(x=-\pi\) and \(x=\pi\). Consider the general form of the tangent function: \(y = \tan(x + k\pi)\). If you add \(k\pi\) to the \(x\) values, the entire graph of the tangent function will shift \(k\pi\) units horizontally but retain the same general shape. This means that if \(k\) is an integer, each of these 'shifted' functions will still pass through the origin and have asymptotes at \(x=-\pi\) and \(x=\pi\).

Key concepts

Periodicity of Tangent Function

When we talk about the periodicity of the tangent function, we refer to its nature of repeating at regular intervals along the x-axis. Mathematically speaking, the tangent function is periodic with a period of \(\pi\), which means that its values repeat every \(\pi\) units.

To visualize this concept, imagine walking in a straight line and seeing the same pattern on the ground after every few steps. Similarly, if you were to trace the tangent function along the x-axis, every segment of \(\pi\) units would look exactly the same as the previous one. This property of the tangent function ensures that it will intersect the origin, \(0,0\), an infinite number of times, each time at an interval of \(\pi\) units.

Graph of Tangent Function

The graph of the tangent function is characterized by its distinct shape, which resembles a series of U and inverted U shapes extending infinitely in both the positive and negative directions along the y-axis. This behavior is due to the fact that the tangent function approaches infinity as the cosine function approaches zero, resulting in what we call 'vertical asymptotes'.

At the origin, the tangent graph starts anew, rising from negative infinity and then falling to positive infinity until it reaches its next asymptote. This pattern is due to the sine and cosine functions, which are the building blocks of the tangent function, as tangent is the ratio of sine over cosine \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). One practical way to plot the graph of the tangent function is to start from the origin, make a curve passing through \(\frac{\pi}{4}, 1\), and continue until you reach the vertical asymptote. Then, repeat this pattern for each period of \(\pi\).

Vertical Asymptotes

Vertical asymptotes are like invisible barriers that the graph of a function can never touch or cross. In the case of the tangent function, we encounter these vertical lines at \(x = \frac{(2n-1)\pi}{2}\) for every integer \(n\). These asymptotes occur exactly where the cosine function equals zero because the tangent function, being the ratio of sine to cosine, approaches infinity or negative infinity at these points.

The presence of vertical asymptotes in the tangent function graph affects its domain, as the function is undefined at these values of \(x\). This is why, when sketching the graph of the tangent function, we draw dashed lines to represent these asymptotes and show the function approaching these lines without ever touching them. Effortlessly understanding the location and significance of these asymptotes is crucial to mastering the behavior of the tangent function and its graph.