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

Verified step by step guidance

1. Identify the basic function and its transformation: The given function is \(y = -2 \tan\left(\frac{1}{4} x\right)\). Here, the basic function is \(\tan(x)\), which has a period of \(\pi\).
2. Determine the period of the transformed tangent function: The period of \(\tan(bx)\) is given by \(\frac{\pi}{|b|}\). In this case, \(b = \frac{1}{4}\), so the period is \(\frac{\pi}{\frac{1}{4}} = 4\pi\).
3. Choose the one-period interval for graphing: Since the period is \(4\pi\), a natural one-period interval to graph over is from \(0\) to \(4\pi\) (or any interval of length \(4\pi\)).
4. Analyze the amplitude and reflection: The coefficient \(-2\) means the graph is vertically stretched by a factor of 2 and reflected across the x-axis. So, the values of \(\tan\left(\frac{1}{4} x\right)\) are multiplied by \(-2\).
5. Identify key points and asymptotes within the interval: The tangent function has vertical asymptotes where its argument equals \(\frac{\pi}{2} + k\pi\), for integers \(k\). Solve \(\frac{1}{4} x = \frac{\pi}{2} + k\pi\) to find asymptotes at \(x = 2\pi + 4k\pi\). Plot these asymptotes and key points (where the function crosses zero) within the chosen interval to sketch the graph.

Key Concepts

Period of the Tangent Function

The period of the basic tangent function y = tan(x) is π. When the function is transformed to y = tan(bx), the period changes to π divided by the absolute value of b. Understanding how to calculate the period is essential for correctly graphing one full cycle of the function.

Amplitude and Vertical Stretch/Compression

Although the tangent function does not have a maximum or minimum amplitude, the coefficient outside the function, such as -2 in y = -2 tan(¼ x), affects the vertical stretch and reflection. A negative coefficient reflects the graph across the x-axis, and the magnitude stretches or compresses the graph vertically.

Asymptotes of the Tangent Function

Tangent functions have vertical asymptotes where the function is undefined, occurring at points where the cosine in the denominator is zero. For y = tan(bx), asymptotes occur at x = (π/2 + kπ)/b for all integers k. Identifying these asymptotes is crucial for accurately sketching the graph.