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The tangent function, denoted by \( \tan x \), is one of the fundamental trigonometric functions. It is defined as the ratio of the sine and cosine functions, meaning \( \tan x = \frac{\sin x}{\cos x} \). This function is unique because of its undefined points, where the cosine of \( x \) is zero. These points, known as asymptotes, cause the tangent graph to approach infinity, leading to its characteristic shape.

Asymptotes typically occur at \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer. Rather than being a line that the graph can touch, an asymptote is a line that the graph infinitely approaches but never actually meets. This creates the repeated pattern where the tangent function "jumps" from \( +\infty \) to \( -\infty \) between each period.

Keep in mind, the tangent function is periodic with a period of \( \pi \), meaning it repeats every \( \pi \) units along the x-axis. Understanding these properties is crucial when working with transformations of the tangent function, like in the exercise where it is scaled by \( \frac{1}{3} \).

Graphing trigonometric functions like the tangent function can initially seem complex, but understanding the essential steps can make it straightforward. Start by considering the basic shape and characteristics of the tangent graph, such as its period and asymptotes.

For the function \( y = \frac{1}{3} \tan x \), the graph has the same period as the standard tangent function, \( \pi \). However, the amplitude is scaled down by \( \frac{1}{3} \). This means that even though the graph approaches infinity near its asymptotes, it does so at a slower rate than the regular tangent graph, giving it its distinct pattern of peaks and valleys.

To graph \( y = \frac{1}{3} \tan x \):

• Identify and mark the asymptotes at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) within the interval from 0 to \( 2\pi \).
• Plot key points where the graph intersects the x-axis, such as \( x = 0, \pi, \) and \( 2\pi \).
• From these key points, sketch the graph moving away towards the asymptotes on both sides, showcasing the scaled amplitude effect.

Following these steps aids in accurately graphing any transformed trigonometric function.

When discussing trigonometric functions, amplitude and period are two fundamental concepts. The amplitude typically measures how "tall" the graph is, indicating the maximum distance the function achieves from the middle line. However, for tangent, sincethe curve can approach infinity, traditional amplitude isn't usually applicable.

Instead, transformations like \( y = \frac{1}{3} \tan x \) alter the slope of the tangent curve, effectively reducing its "vertical reach" without impacting the infinite nature. This factor represents how the curve's steepness has changed. As a result, each curve appears more compressed vertically, compared to \( y = \tan x \).

The period of a trigonometric function indicates the length needed for the function to start repeating its pattern. For both the tangent and its scaled form \( y = \frac{1}{3} \tan x \), the period remains \( \pi \), indicating that despite vertical transformations, the horizontal length over which the curve repeats doesn't change.

• Amplitude for cosine and sine measures the oscillation height; for tangent, it involves slope steepness.
• The period is consistent over horizontal transformations, essential for predicting graph behavior.

Understanding amplitude and period is essential for accurately interpreting and graphing trigonometric functions.