Graph each function over a one-period interval.
y = 2 tan (¼ x)

Verified step by step guidance

1. Identify the basic form of the tangent function: \( y = a \tan(bx) \). In this case, \( a = 2 \) and \( b = \frac{1}{4} \).
2. Determine the period of the tangent function, which is given by \( \frac{\pi}{b} \). Substitute \( b = \frac{1}{4} \) to find the period.
3. Calculate the period: \( \text{Period} = \frac{\pi}{\frac{1}{4}} = 4\pi \). This is the interval over which you will graph one complete cycle of the function.
4. Identify the vertical stretch factor, which is \( a = 2 \). This means the graph will be stretched vertically by a factor of 2 compared to the standard tangent function.
5. Sketch the graph over the interval \( [0, 4\pi] \), noting the vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) for integers \( n \), and plot key points such as the origin and the points where the function crosses the x-axis.

Key Concepts

Period of a Trigonometric Function

The period of a trigonometric function is the length of one complete cycle of the function. For the tangent function, the standard period is π. However, when the function is modified, such as in y = 2 tan(¼ x), the period changes based on the coefficient of x. In this case, the period becomes 4π, as it is calculated by dividing the standard period by the coefficient of x.

Transformation of Functions

Transformations of functions involve shifting, stretching, or compressing the graph of a function. In the function y = 2 tan(¼ x), the '2' indicates a vertical stretch, meaning the output values of the tangent function are multiplied by 2, making the graph steeper. The '¼' indicates a horizontal stretch, which affects the period of the function, resulting in a wider graph.

Graphing the Tangent Function

The tangent function has unique characteristics, including vertical asymptotes where the function is undefined, occurring at odd multiples of π/2. When graphing y = 2 tan(¼ x), it is essential to identify these asymptotes and the points where the function crosses the x-axis. The graph will repeat every 4π, and understanding the behavior of the tangent function helps in accurately plotting its graph over the specified interval.