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When it comes to understanding the period of a tangent function, one should know that it is quite different from sine or cosine functions.
The standard tangent function has a period of \( \pi \). This is because, unlike sine and cosine functions that repeat every \( 2\pi \), tangent functions complete their cycle when the tangent value returns to its original state after \( \pi \) radians.
The period can be altered by changing the coefficient before the variable \( x \) inside the tangent function. If the function is \( \tan(nx) \), the period becomes \( \frac{\pi}{n} \).
In our exercise, the function is \( y = 2\tan(2x) \).

• The coefficient is 2, thus the period becomes \( \frac{\pi}{2} \).

This means that the function repeats itself every \( \frac{\pi}{2} \) radians, compressing the graph compared to the base function.

Amplitude, a common concept with sine and cosine functions, does not apply to tangent functions in the same way.
The reason is that tangent functions extend infinitely in both vertical directions.
This is why the amplitude of a tangent function is considered \( \infty \). It's the range of output values, stretching from \( -\infty \) to \( +\infty \).

• In the case of \( y = 2\tan(2x) \), the multiplier of 2 affects the steepness of the graph rather than its amplitude.

You might see the function growing faster, but it never limits or corrects the extent of outputs, thus still remaining with no fixed amplitude.

Vertical asymptotes in tangent functions are crucial as they indicate where the function is undefined.
The tangent function is undefined when \( \cos(x) = 0 \), which occurs at odd multiples of \( \frac{\pi}{2} \).
For the function \( y = 2\tan(2x) \), the factor inside the tangent alters the location of these asymptotes.

• Here, the asymptotes will occur at odd multiples of \( \frac{\pi}{4} \).

Asymptotes guide us regarding how the graph approaches these undefined points, diving towards\( -\infty \) on one side and rising to \( +\infty \) on the other.

Phase shift in trigonometric functions dictates the horizontal displacement of the graph along the \( x \)-axis.
In basic terms, it tells us if the wave has moved left or right from its standard position.
For tangent functions like \( y = \tan(x + c) \), where \( c \) is a constant, the graph shifts horizontally by \( \frac{-c}{n} \) units, where \( n \) is the coefficient of \( x \).
In our case of \( y = 2\tan(2x) \), there's no constant added or subtracted from \( x \), resulting in no phase shift.

• The function \( y = 2\tan(2x) \) starts its regular cycle at \( x = 0 \).

This further illustrates how the tangent graph in this instance remains perfectly aligned from its origin, with no horizontal alterations necessary.