Introduction

In trigonometry and precalculus, the tangent function plays a key role. It helps us understand angles and their relationships. The focus here will be on the characteristics of the tangent function graph, especially its period and asymptotes. By delving into these features, students can improve their understanding and excel in the AP® Precalculus exam.

Understanding the Tangent Function

The tangent function is defined as the ratio of the sine to the cosine function:

\text{Tangent Function: } \tan(x) = \frac{\sin(x)}{\cos(x)}

What’s special about the tangent function? Well, unlike the sine and cosine functions, which are waves, the tangent function does not have a horizontal maximum or minimum. Instead, it continues infinitely in the vertical direction.

Key Characteristics of the Tangent Function

Period of Tangent Function

The period of a function is the length of one complete cycle before it repeats itself. For the tangent function, this period is \pi. This means after an interval of \pi, the tangent graph repeats itself.

Example: Graphing One Full Period of a Tangent Function

1. Identify the period: \pi.
2. Plot critical points: Choose five points in between two consecutive asymptotes. Typically, we start by looking between x= -\pi/2 and x=\pi/2
   • (-\pi/3, -\sqrt{3})
   • (-\pi/4, -1)
   • (0, 0)
   • (\pi/4, 1)
   • (\pi/3, \sqrt{3})
3. Draw the curve: Recognizing the “S” shape, starting from negative infinity and increasing towards positive infinity in between two consecutive asymptotes.

Tangent Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. The tangent function has vertical asymptotes because \tan(x) becomes undefined where \cos(x) = 0. These occur at:

x = \frac{\pi}{2} + k\pi

…where (k) is an integer.

Example: Identifying Asymptotes in a Tangent Function Graph

1. Calculate asymptotes: For the first period, asymptotes at x = \frac{\pi}{2} and x = -\frac{\pi}{2}.
2. Mark these on the graph as dashed vertical lines.

Behavior Between Asymptotes

Between its asymptotes, the graph of the tangent function transitions smoothly, resembling an “S” curve. As x increases from one asymptote to the next, the graph transitions from concave down to concave up, crossing through the x-axis.

Example: Analyzing the Graph of Tangent Within One Period

1. Identify points of inflection: x = 0 results in y = 0.
2. Observe the curve’s direction: As x approaches \pi/2, the graph curves upward rapidly.

Sketching the Graph of the Tangent Function

To sketch the tangent function graph:

1. Identify the period: \pi.
2. Locate asymptotes: Place vertical lines at x = \frac{\pi}{2} and x = -\frac{\pi}{2}. You can plot more by adding multiples of \pi .
3. Plot key points: (-\pi/4, -1), (0,0), (\pi/4, 1)
4. Sketch a smooth “S” shape, expanding towards the asymptotes.
5. Check for symmetry: The curve should be symmetrical around the origin for each segment.

Quick Reference Chart of Key Vocabulary

Conclusion

In summary, understanding the tangent function involves recognizing its period of \pi and identifying its vertical asymptotes. The transition from concave down to concave up creates the unique “S” shape within one full period of tangent. By practicing sketching and analyzing this function, students can improve their problem-solving skills and effectively prepare for the AP® Precalculus exam. Happy graphing!

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