In Exercises 18–24, graph two full periods of the given tangent or cotangent function. y = −2 tan π/4 x

Verified step by step guidance

1. Identify the given function: \(y = -2 \tan\left(\frac{\pi}{4} x\right)\). This is a tangent function with a vertical stretch and a horizontal compression.
2. Recall the general form of the tangent function: \(y = a \tan(bx)\), where \(a\) affects the amplitude (vertical stretch) and \(b\) affects the period (horizontal stretch/compression).
3. Calculate the period of the function using the formula for tangent: \(\text{Period} = \frac{\pi}{|b|}\). Here, \(b = \frac{\pi}{4}\), so the period is \(\frac{\pi}{\frac{\pi}{4}} = 4\).
4. Since the problem asks for two full periods, determine the interval for \(x\) to graph: from \(0\) to \(8\) (because \(2 \times 4 = 8\)).
5. Plot key points and asymptotes for the tangent function within the interval \([0,8]\). The vertical asymptotes occur where the argument of tangent is \(\frac{\pi}{2} + k\pi\), so solve \(\frac{\pi}{4} x = \frac{\pi}{2} + k\pi\) for \(x\) to find asymptotes, then sketch the curve with vertical stretch \(-2\) (which flips and stretches the graph vertically).

Key Concepts

Period of 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/|b|. Understanding how to calculate the period is essential for graphing two full periods of the function accurately.

Amplitude and Vertical Stretch

Although tangent functions do not have a maximum amplitude, the coefficient outside the function, such as -2 in y = -2 tan(π/4 x), affects the vertical stretch and reflection. The negative sign reflects the graph across the x-axis, and the magnitude stretches the graph vertically by a factor of 2.

Asymptotes of Tangent Function

Tangent functions have vertical asymptotes where the function is undefined, occurring at x-values where the cosine is zero. For y = tan(bx), asymptotes occur at x = (π/2 + kπ)/b. Identifying these asymptotes is crucial for sketching the graph correctly.