91Ó°ÊÓ

Understanding the period of a tangent function is crucial for graphing it correctly. The period refers to the horizontal length of one full cycle of the graph. For standard tangent functions, the period is \(\pi\), meaning the pattern of the graph repeats every \(\pi\) units on the x-axis.

The function \(y = \frac{1}{3}\tan x\) maintains the same period as the standard tangent function because it is not horizontally stretched or compressed. Thus, to sketch two full periods of this function, one would span the x-axis from \(0\) to \(2\pi\). This characteristic is essential to keep in mind, as it remains a foundational aspect when graphing any variant of the tangent function.

Asymptotes are vertical lines that the graph of a function approaches but never touches or crosses. For a standard tangent function, asymptotes occur at \(x = \pm\frac{\pi}{2} + n\pi\), where \(n\) is an integer. In the example \(y = \frac{1}{3}\tan x\), since there are no transformations affecting the location of the asymptotes, they remain at the same positions as the standard function.

Remembering that asymptotes indicate 'boundaries' where the function's output is undefined can help you correctly draw the tangent curve, which sharply increases or decreases as it approaches these lines. When graphing, these asymptotes should be carefully plotted, as they dictate the behavior of the graph between the peaks and valleys of the tangent curve.

Sketching tangent graphs requires a systematic approach. Begin by marking the asymptotes on your coordinate plane, as they guide the overall structure of the graph. For the function \(y = \frac{1}{3}\tan x\), draw vertical dashed lines at \(x = \pm\frac{\pi}{2}\) and \(x = \pm\frac{3\pi}{2}\), etc. With the asymptotes in place, plot the origin at (0,0), assuming the graph has not been shifted.

Then, draw the curve starting at the origin and proceed toward the asymptotes. Since the graph has a multiplicative factor of \(\frac{1}{3}\), the rise over the run of the graph will be less steep compared to the standard tangent curve. After sketching one period, ensure continuity by starting the next period at the end of the previous one, maintaining the same slope. The graph will be symmetrical about the origin, so the second and fourth quadrants are reflections of the first and third quadrants, respectively.

Transformations of tangent functions involve shifting, stretching, or compressing the basic tangent curve. These transformations can be identified by changes in the function's equation relative to the standard form \(y = \tan x\). In the function \(y = \frac{1}{3}\tan x\), there's a vertical compression represented by the coefficient \(\frac{1}{3}\) which affects the 'steepness' of the graph.

Other potential transformations include horizontal shifts (left or right), vertical shifts (up or down), horizontal stretches or compressions (affecting the period), and reflections (inverting the graph across an axis). Understanding these transformations is key to accurately plotting any variant of the tangent function. Always start by identifying these transformations before graphing to avoid common mistakes.