Graph each function over a one-period interval. See Examples 1–3.
y = 2 tan x

Verified step by step guidance

1. Identify the basic period of the tangent function. The function \( \tan x \) has a period of \( \pi \), so the function \( y = 2 \tan x \) will also have a period of \( \pi \).
2. Determine the one-period interval to graph. Since the period is \( \pi \), a common choice is to graph from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \), where the tangent function has vertical asymptotes at the endpoints.
3. Note the vertical asymptotes of \( y = 2 \tan x \) occur where \( \tan x \) is undefined, which is at \( x = \pm \frac{\pi}{2} \). Draw dashed vertical lines at these points to indicate the asymptotes.
4. Plot key points within the interval. For example, at \( x = 0 \), \( y = 2 \tan 0 = 0 \). At \( x = \frac{\pi}{4} \), calculate \( y = 2 \tan \frac{\pi}{4} = 2 \times 1 = 2 \). Similarly, at \( x = -\frac{\pi}{4} \), \( y = 2 \times (-1) = -2 \).
5. Sketch the curve between the asymptotes using the plotted points, remembering that the tangent function increases without bound near the asymptotes and passes through the origin with the shape stretched vertically by a factor of 2.

Key Concepts

Period of the Tangent Function

The tangent function has a fundamental period of π, meaning its values repeat every π units along the x-axis. When graphing y = 2 tan x, it is essential to plot the function over one full period, typically from -π/2 to π/2, where the function is continuous and exhibits its characteristic shape.

Vertical Asymptotes of Tangent

Tangent has vertical asymptotes where the function is undefined, occurring at x = ±π/2, ±3π/2, etc. These asymptotes represent values where the function approaches infinity or negative infinity, and they are critical for accurately sketching the graph of y = 2 tan x.

Amplitude and Vertical Stretch

Although tangent does not have a maximum or minimum amplitude, the coefficient 2 in y = 2 tan x vertically stretches the graph, making the function values grow twice as fast. This affects the steepness of the curve but does not change the period or location of asymptotes.