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, y = −tan(x − π/2).

Verified step by step guidance

1. Identify the vertical asymptotes of the tangent graph. From the graph, the asymptotes are at \(x = -\frac{3\pi}{4}\) and \(x = \frac{\pi}{4}\).
2. Recall that the standard tangent function \(y = \tan x\) has vertical asymptotes at \(x = \pm \frac{\pi}{2}\), and the period is \(\pi\). The asymptotes in the given graph are shifted horizontally compared to the standard tangent function.
3. Determine the horizontal shift by comparing the asymptotes of the given graph to those of \(y = \tan x\). The asymptotes of \(y = \tan x\) are at \(x = \pm \frac{\pi}{2}\), but here they are at \(x = -\frac{3\pi}{4}\) and \(x = \frac{\pi}{4}\). This indicates a shift of \(-\frac{\pi}{4}\) to the left.
4. Use the horizontal shift to write the equation of the tangent function. A shift to the left by \(\frac{\pi}{4}\) means the function is \(y = \tan\left(x + \frac{\pi}{4}\right)\). However, this is not one of the options, so check if the function is reflected or shifted further.
5. Check the zeros of the function at \(x = -\frac{\pi}{4}\) and \(x = \frac{3\pi}{4}\), and compare with the options. By analyzing the behavior and the given options, conclude which equation matches the graph based on the shifts and reflections.

Key Concepts

Tangent Function and Its Graph

The tangent function, y = tan(x), has vertical asymptotes where the cosine function is zero, at x = ±π/2, ±3π/2, etc. Its graph repeats every π units, showing periodic behavior. Understanding the shape and asymptotes of the tangent graph is essential for identifying transformations.

Horizontal Shifts of Trigonometric Functions

A horizontal shift in the function y = tan(x + c) moves the graph left or right by c units. This shift changes the location of vertical asymptotes and zeros accordingly. Recognizing how shifts affect the graph helps match equations to their graphs.

Reflection of the Tangent Function

Multiplying the tangent function by -1, as in y = -tan(x), reflects the graph across the x-axis. This changes the direction of the curve between asymptotes but does not affect the location of asymptotes. Identifying reflections is key to distinguishing between similar tangent graphs.