For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = tan 3x

Verified step by step guidance

1. Identify the standard form of the tangent function: $y=a\tan (bx-c)+d$.
2. Compare the given function $y=\tan (3x)$ with the standard form to identify parameters: $a=1$, $b=3$, $c=0$, and $d=0$.
3. Determine the amplitude: The tangent function does not have an amplitude because it has no maximum or minimum values.
4. Calculate the period: The period of the tangent function is given by $\frac{\pi }{b}$. Substitute $b=3$ to find the period.
5. Identify the vertical translation and phase shift: Since $d=0$, there is no vertical translation. Since $c=0$, there is no phase shift.

Key Concepts

Amplitude

Amplitude refers to the maximum extent of a periodic function from its central axis. For trigonometric functions like sine and cosine, amplitude is a key characteristic, but it does not apply to the tangent function. Therefore, when analyzing the function y = tan(3x), we note that it does not have an amplitude since it can take on all real values.

Period

The period of a trigonometric function is the length of one complete cycle of the function. For the tangent function, the standard period is π. However, when the function is modified, such as in y = tan(3x), the period is adjusted by the coefficient of x. Specifically, the period becomes π/3, indicating that the function completes one full cycle over this interval.

Phase Shift

Phase shift refers to the horizontal displacement of a periodic function from its standard position. For functions of the form y = tan(bx - c), the phase shift can be calculated as c/b. In the case of y = tan(3x), there is no horizontal shift since there is no constant added or subtracted from the argument of the tangent function, resulting in a phase shift of zero.