Problem 51 - A Graphical Approach to Precalculus with Limits

Question

Graph each function over a one-period interval. $$y=\tan 4 x$$

Step-by-Step Solution

Answer

The function \( y = \tan 4x \) completes one period over \( 0 \) to \( \frac{\pi}{4} \) with a vertical asymptote at \( \frac{\pi}{8} \).

Step 1: Determine the Period of the Function

The period of a tangent function is given by \( \pi \) divided by the coefficient of \(x\) inside the tangent function. The function given is \( y = \tan(4x) \), so the period is: \[ \text{Period} = \frac{\pi}{4} \]. This means one complete cycle of \( y = \tan(4x) \) is completed over an interval of \( \frac{\pi}{4} \).

Step 2: Identify Key Points within One Period

For tangent functions, key points occur at intervals of \( \frac{\text{Period}}{2} \), \( \frac{\text{Period}}{4} \), and \( \frac{3\text{Period}}{4} \). Thus, the key points for \( y = \tan(4x) \) within one period are: \( 0 \), \( \frac{\pi}{8} \), and \( \frac{3\pi}{8} \). At these points, the function's values are: \( \tan(0) = 0 \), vertical asymptote at \( \frac{\pi}{8} \), and \( \tan(\frac{3\pi}{8}) \) which approaches infinity as it nears \( \frac{\pi}{4} \).

Step 3: Plot the Graph

Using the period \( \frac{\pi}{4} \) determined in Step 1 and the key points from Step 2, plot the graph. Start with \( y=0 \) at \( x=0 \), progress to a vertical asymptote at \( x = \frac{\pi}{8} \), passing through the origin gain before completing the cycle. The graph shall rise steeply as it approaches \( \frac{\pi}{8} \), reflect at zero, and continue growing steeply until it completes the period. Repeat this pattern within \( 0 \) and \( \frac{\pi}{4} \).

Key Concepts

Tangent Function

The tangent function, denoted as \( \tan(x) \), is one of the fundamental trigonometric functions. It relates the angle \( x \) in a right triangle to the ratio of the opposite side over the adjacent side.

• The basic properties of the tangent function include its periodicity and undefined points.
• Unlike the sine and cosine functions, \( \tan(x) \) has vertical asymptotes where the function is undefined. These occur at odd multiples of \( \frac{\pi}{2} \).
• The range of the tangent function is all real numbers, going from \(-\infty\) to \(\infty\).

Tangent is important for understanding how angles relate to side ratios, especially in various applications such as physics and engineering. When graphing \( \tan(x) \), note that it crosses the x-axis every \( n\pi \), where \( n \) is an integer.

Trigonometric Graphing

Graphing trigonometric functions is a skill that involves understanding their periodic nature and identifying key points within one complete cycle. To graph \( y = \tan(4x) \), you must focus on:

• Determining the period: For tangent, this is affected by the coefficient of \( x \). Here, it compresses the period to \( \frac{\pi}{4} \).
• Plotting key points: Identify symmetry, zero crossings, and asymptotes. For \( \tan(4x) \), key points include the origin, \( \frac{\pi}{8} \), and reflect near \( \frac{3\pi}{8} \).

Graphing effectively requires patience and a keen sense of symmetry and periodic patterns. Start plotting near zero and follow through asymptotes, ensuring each segment reflects the function's rise and fall behavior.

Function Periodicity

Periodicity is a vital concept in the context of trigonometric functions. It refers to the repeating nature of these functions. For tangent functions like \( y = \tan(4x) \), the period signifies the interval length over which the graph repeats itself.

• This is calculated as \( \frac{\pi}{b} \), where \( b \) is the coefficient of \( x \). So a \( \tan(4x) \) function has a period of \( \frac{\pi}{4} \).
• Understanding periodicity enables prediction of function behavior over different intervals, crucial for plotting accurate graphs.
• Because of its periodicity, once you have one period, the patterns repeat ad infinitum in both directions along the x-axis.

This rhythmic quality of the tangent function showcases the cyclic nature inherent in all trigonometric functions, making them predictable and manageable to graph.