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

Verified step by step guidance

1. Step 1: Understand the basic properties of the tangent function, $\tan x$, which has a period of $\pi$. This means the function repeats every $\pi$ units.
2. Step 2: Recognize that the function $y=1+\tan x$ is a vertical shift of the basic tangent function by 1 unit upwards.
3. Step 3: Determine the period of the function $y=1+\tan x$. Since the period of $\tan x$ is $\pi$, the period of $y=1+\tan x$ remains $\pi$.
4. Step 4: Identify the two-period interval for graphing. Since the period is $\pi$, a two-period interval would be from $0$ to $2\pi$.
5. Step 5: Plot the function over the interval $0$ to $2\pi$, taking into account the vertical shift. The graph will have vertical asymptotes at $x=\frac{\pi }{2}+n\pi$ for integers $n$, and the function will pass through points like $(0,1)$, $(\pi ,1)$, etc.

Key Concepts

Tangent Function

The tangent function, denoted as tan(x), is a periodic function defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). It has a period of π, meaning it repeats its values every π radians. The function has vertical asymptotes where the cosine function equals zero, specifically at x = (π/2) + nπ, where n is any integer.

Graphing Periodic Functions

Graphing periodic functions involves plotting the function over a specified interval to visualize its repeating nature. For the tangent function, one must consider its asymptotes and the points where it crosses the x-axis. In this case, the function y = 1 + tan(x) shifts the entire graph of tan(x) upward by 1 unit, affecting its intercepts and vertical asymptotes.

Transformations of Functions

Transformations of functions involve shifting, stretching, or reflecting the graph of a function. In the case of y = 1 + tan(x), the '+1' indicates a vertical shift upwards by one unit. Understanding transformations is crucial for accurately graphing functions, as they alter the position and shape of the original graph while maintaining its overall periodicity.