In Exercises 1–4, the graph of a tangent function is given. Select the equation for each graph from the following options:y = tan(x + π/2), y = tan(x + π), y = −tan(x − π/2).

The tangent function, defined as tan(x) = sin(x)/cos(x), has a periodicity of π, meaning it repeats every π units. It also has vertical asymptotes where the cosine function equals zero, specifically at x = (π/2) + nπ for any integer n. Understanding these properties is crucial for analyzing the behavior of tangent graphs.

Phase shift refers to the horizontal translation of a trigonometric function. For the tangent function, an equation of the form y = tan(x + c) indicates a shift to the left by c units, while y = tan(x - c) indicates a shift to the right. This concept is essential for determining how the graph of the tangent function is altered by changes in its equation.

Vertical asymptotes in the graph of the tangent function occur at points where the function is undefined, specifically where the cosine of the angle is zero. These asymptotes indicate the boundaries of the function's range and help in identifying the intervals where the function increases or decreases. Recognizing the locations of these asymptotes is vital for accurately sketching or interpreting tangent graphs.