Graph each function over a two-period interval.
y = -1 + 2 tan x

Verified step by step guidance

1. Identify the base function and its characteristics: The given function is \(y = -1 + 2 \tan x\). The base function here is \(\tan x\), which has a period of \(\pi\).
2. Determine the interval for graphing: Since the problem asks for a two-period interval, and the period of \(\tan x\) is \(\pi\), the interval will be \([0, 2\pi]\) or any other interval of length \(2\pi\).
3. Analyze the transformations applied to the base function: The function has a vertical stretch by a factor of 2 (due to the coefficient 2 in front of \(\tan x\)) and a vertical shift downward by 1 unit (due to the \(-1\)).
4. Identify the vertical asymptotes of the function: The vertical asymptotes of \(\tan x\) occur at \(x = \frac{\pi}{2} + k\pi\), where \(k\) is any integer. These asymptotes will remain the same for the transformed function.
5. Sketch the graph by plotting key points and asymptotes over the two-period interval, applying the vertical stretch and shift to the \(\tan x\) values, and marking the asymptotes at the appropriate \(x\)-values.

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 over a two-period interval, you plot the function from an initial point to two times π, capturing two full cycles of the tangent curve.

Vertical Asymptotes of Tangent

Tangent has vertical asymptotes where the function is undefined, occurring at x = (2k+1)π/2 for all integers k. These asymptotes divide the graph into sections and indicate where the function approaches infinity or negative infinity, crucial for accurate graphing.

Transformations of Trigonometric Functions

The function y = -1 + 2 tan x involves vertical scaling by 2 and a vertical shift downward by 1. Scaling affects the steepness of the graph, while shifting moves the entire curve down, altering the midline and range but not the period or asymptotes.