For each function, give the amplitude, period, vertical translation, and phase shift, as applicable.
y = 1/3 tan (3x - π/3)

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 (though tangent functions do not have a traditional amplitude), \(B\) affects the period, \(C\) affects the phase shift, and \(D\) is the vertical translation.
2. Determine the amplitude: For tangent functions, amplitude is not defined because the function's values range from \(-\infty\) to \(+\infty\). So, amplitude is not applicable here.
3. Calculate the period using the formula \(\text{Period} = \frac{\pi}{|B|}\). Here, \(B = 3\), so the period is \(\frac{\pi}{3}\).
4. Find the phase shift using the formula \(\text{Phase shift} = \frac{C}{B}\). Given the expression inside the tangent is \(3x - \frac{\pi}{3}\), \(C = \frac{\pi}{3}\), so phase shift is \(\frac{\pi/3}{3} = \frac{\pi}{9}\). Since it is \(3x - \frac{\pi}{3}\), the phase shift is to the right by \(\frac{\pi}{9}\).
5. Identify the vertical translation \(D\): Since there is no constant added or subtracted outside the tangent function, the vertical translation 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 extend infinitely.

Period of the Tangent Function

The period of a tangent function is the length of one complete cycle before it repeats. The standard period of tan(x) is π, and it changes with the coefficient inside the function as period = π / |b|, where b is the coefficient of x. This affects how frequently the function repeats its pattern.

Phase Shift and Vertical Translation

Phase shift refers to the horizontal shift of the graph, calculated by solving the inside of the function for zero (bx - c = 0). Vertical translation is the upward or downward shift of the graph, determined by any constant added or subtracted outside the function. Both shifts affect the position but not the shape of the graph.