Graph each function over a one-period interval. See Examples 1–3.
y = tan 4x

Verified step by step guidance

1. Identify the function given: \(y = \tan 4x\). This is a tangent function with an argument multiplied by 4, which affects its period.
2. Recall the period formula for the tangent function: the standard period of \(\tan x\) is \(\pi\). For \(y = \tan(bx)\), the period is \(\frac{\pi}{b}\). Here, \(b = 4\), so the period is \(\frac{\pi}{4}\).
3. Determine the one-period interval for the function. Since the period is \(\frac{\pi}{4}\), you can choose an interval of length \(\frac{\pi}{4}\), for example from \(0\) to \(\frac{\pi}{4}\) or from \(-\frac{\pi}{8}\) to \(\frac{\pi}{8}\).
4. Identify the vertical asymptotes within this interval. For \(y = \tan 4x\), vertical asymptotes occur where the argument \(4x = \frac{\pi}{2} + k\pi\), for any integer \(k\). Solve for \(x\) to find asymptotes: \(x = \frac{\pi}{8} + \frac{k\pi}{4}\).
5. Plot key points within the chosen interval, including where \(y=0\) (which occurs when \(4x = k\pi\), so \(x = \frac{k\pi}{4}\)), and sketch the curve approaching the asymptotes, showing the typical shape of the tangent function within one period.

Key Concepts

Period of the Tangent Function

The period of the basic tangent function y = tan(x) is π. When the function is transformed to y = tan(bx), the period changes to π divided by the absolute value of b. For y = tan(4x), the period is π/4, meaning the function repeats every π/4 units along the x-axis.

Graphing Tangent Functions

Graphing tangent functions involves identifying key points such as zeros, asymptotes, and the shape between them. Tangent has vertical asymptotes where the function is undefined, occurring at x-values where the cosine is zero. For y = tan(4x), asymptotes occur at x = (2k+1)π/8, where k is an integer.

One-Period Interval

A one-period interval is the length along the x-axis over which the function completes one full cycle before repeating. For y = tan(4x), this interval is from 0 to π/4 or any interval of length π/4. Graphing over one period helps visualize the function's behavior without redundancy.