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 general form of the tangent function: \(y = a \tan(bx - c) + d\), where \(a\) is the amplitude, \(b\) affects the period, \(c\) is the phase shift, and \(d\) is the vertical translation.
2. Note that the tangent function does not have an amplitude because its values range from \(-\infty\) to \(+\infty\), so amplitude is not defined for \(y = \tan 3x\).
3. Determine the period using the formula for tangent: \(\text{Period} = \frac{\pi}{|b|}\). Here, \(b = 3\), so the period is \(\frac{\pi}{3}\).
4. Check for vertical translation \(d\). Since there is no constant added or subtracted outside the tangent function, the vertical translation is \(0\).
5. Find the phase shift using \(\frac{c}{b}\). Since the function is \(y = \tan 3x\) with no subtraction inside the argument, \(c = 0\), so the phase shift is \(0\).

Key Concepts

Amplitude of Trigonometric Functions

Amplitude measures the maximum vertical distance from the midline to the peak of a wave. For sine and cosine functions, amplitude is the absolute value of the coefficient before the function. However, tangent functions do not have an amplitude because their values increase without bound.

Period of the Tangent Function

The period of a function is the length of one complete cycle on the x-axis. The basic tangent function, y = tan x, has a period of π. When the function is y = tan(bx), the period changes to π divided by the absolute value of b, compressing or stretching the graph horizontally.

Phase Shift and Vertical Translation

Phase shift refers to the horizontal shift of the graph, determined by any added or subtracted value inside the function's argument. Vertical translation shifts the graph up or down, determined by any constant added or subtracted outside the function. In y = tan 3x, there is no phase shift or vertical translation since no such terms are present.