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).

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1. Identify the key features of the tangent graph, such as the vertical asymptotes and x-intercepts.
2. Notice that the vertical asymptotes are at x = -3π/4 and x = π/4, which suggests a horizontal shift.
3. The x-intercepts are at (-π/4, 0) and (3π/4, 0), indicating the graph is shifted horizontally.
4. Compare these features with the given equations: y = tan(x + π/2), y = tan(x + π), and y = −tan(x − π/2).
5. Determine which equation matches the observed shifts and asymptotes of the graph.

Key Concepts

Tangent Function Properties

The tangent function, defined as tan(x) = sin(x)/cos(x), has a periodic nature with a period of π. It exhibits 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

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 leftward shift by c units, while y = tan(x - c) indicates a rightward shift. This concept is essential for determining how the graph of the tangent function is altered by changes in its equation.

Vertical Asymptotes

Vertical asymptotes in the graph of the tangent function occur at points where the function is undefined, specifically where cos(x) = 0. In the provided graph, the vertical asymptotes at x = -π/4 and x = π/4 indicate the boundaries of the function's behavior, where it approaches infinity. Identifying these asymptotes helps in selecting the correct equation for the graph.