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).

Verified step by step guidance

1. Identify the key features of the tangent graph, such as the x-intercepts and vertical asymptotes.
2. Notice that the graph has x-intercepts at \(x = -\frac{2\pi}{3}\) and \(x = \frac{\pi}{3}\).
3. Observe the vertical asymptotes at \(x = -\frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\).
4. Compare these features with the standard tangent function \(y = \tan(x)\), which has x-intercepts at integer multiples of \(\pi\) and vertical asymptotes at odd multiples of \(\frac{\pi}{2}\).
5. Determine which transformation (horizontal shift and/or reflection) matches the given graph to one of the provided equations.

Key Concepts

Tangent Function Properties

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

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

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.