In Exercises 45–52, graph two periods of each function.y = 2 tan(x − π/6) + 1

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1. Identify the basic form of the tangent function: \( y = a \tan(b(x - c)) + d \). In this case, \( a = 2 \), \( b = 1 \), \( c = \frac{\pi}{6} \), and \( d = 1 \).
2. Determine the period of the function. The period of \( \tan(bx) \) is \( \frac{\pi}{b} \). Here, \( b = 1 \), so the period is \( \pi \).
3. Identify the phase shift, which is determined by \( c \). The function is shifted to the right by \( \frac{\pi}{6} \).
4. Determine the vertical shift, which is \( d = 1 \). This means the entire graph is shifted up by 1 unit.
5. Graph the function by plotting key points over two periods, considering the vertical stretch (\( a = 2 \)), phase shift, and vertical shift. Start from the phase shift and plot points at intervals of \( \frac{\pi}{4} \) to capture the behavior of the tangent function.

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 units. Understanding the properties of the tangent function, including its asymptotes and behavior near these points, is crucial for graphing it accurately.

Transformations of Functions

Transformations involve shifting, stretching, or compressing the graph of a function. In the given function y = 2 tan(x − π/6) + 1, the term (x − π/6) represents a horizontal shift to the right by π/6, while the coefficient 2 indicates a vertical stretch by a factor of 2. The +1 shifts the entire graph upward by 1 unit, affecting the function's range and position.

Graphing Periodic Functions

Graphing periodic functions requires understanding their key features, such as amplitude, period, phase shift, and vertical shift. For the tangent function, identifying the asymptotes, which occur where the function is undefined, is essential. By plotting key points and considering the transformations, one can accurately represent two periods of the function on a graph.