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The period of a trigonometric function like the tangent gives us information about how often the function repeats its pattern. For a standard tangent function, \( y = \tan x \), the period is \( \pi \). This means that the function repeats itself every \( \pi \) units along the x-axis.

However, when the function is modified, as in \( y = \tan \frac{x}{3} \), changes to the period occur. The effect of dividing \( x \) by 3 is a reduction in frequency. Therefore, each cycle of the tangent function takes longer to complete. The new period becomes \( 3\pi \) in this case.

Understanding how the period is determined is crucial because it affects the entire graph's appearance and the function's repetition interval. To find the new period, remember this key formula for tangent functions: \[ \text{New Period} = \text{Original Period} \times \text{Frequency Factor} \] Therefore, \( 3\pi = \pi \times 3 \).

This shows how dividing the input \( x \) by a number affects the period: it is extended by the same factor.

Vertical asymptotes are lines where the function is undefined. For the tangent function, these occur whenever the tangent approaches infinity.

In the base tangent function \( y = \tan x \), vertical asymptotes are located at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer. Essentially, these are points where the tangent function tries to divide by zero in its typical sinusoidal behavior, causing it to trend toward infinity.

When looking at \( y = \tan \frac{x}{3} \), the placement of these asymptotes shifts due to the interaction of the period change. Specifically, the input \( \frac{x}{3} = \frac{\pi}{2} + n\pi \) gives \( x = \frac{3\pi}{2} + 3n\pi \), thanks to the multiplication factor. This tells us that the vertical asymptotes in the graph of \( y = \tan \frac{x}{3} \) are at every multiple of \( 3\pi \).

Understanding where these vertical asymptotes lie helps in sketching and analyzing the trigonometric function across its domain.

Observing the behavior of the tangent function can provide crucial insights into its graphing and pattern. As \( x \) increases within each period cycle, the tangent function exhibits a distinct behavior.

For \( y = \tan \frac{x}{3} \), it behaves as follows: it starts the period by falling from \(-\infty\), crosses through zero, and continues to rise to \(+\infty\) as \( x \) approaches the next vertical asymptote at \( 3\pi \).

This behavior is consistent across each full period of \( 3\pi \), making it predictable and systematic. The function climbs from negative infinity to positive infinity, giving it its characteristic repeating S-shape every \( 3\pi \).

Recognizing this pattern helps in understanding how tangent behaves and what to expect across its cycle, crucial for both plotting graphs and understanding functional transformations.