Graph two periods of the given tangent function. y = −2 tan (1/2) x

Verified step by step guidance

1. Identify the general form of the tangent function: \(y = a \tan(bx)\), where \(a\) affects the amplitude (vertical stretch) and \(b\) affects the period of the function.
2. Determine the period of the function using the formula for tangent: \(\text{Period} = \frac{\pi}{|b|}\). Here, \(b = \frac{1}{2}\), so calculate the period as \(\frac{\pi}{\frac{1}{2}}\).
3. Calculate the period explicitly: \(\frac{\pi}{\frac{1}{2}} = 2\pi\). This means one full cycle of the tangent function occurs over an interval of length \(2\pi\).
4. Since the problem asks for two periods, plan to graph the function over an interval of length \(2 \times 2\pi = 4\pi\). This will cover two full cycles of the tangent function.
5. Note the vertical stretch and reflection: the coefficient \(-2\) means the graph is vertically stretched by a factor of 2 and reflected across the x-axis. Use this to sketch the graph accordingly, marking key points such as vertical asymptotes where the tangent function is undefined.

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 this helps determine the length of one full cycle to graph two periods 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(½x), vertically stretches the graph by a factor of 2 and reflects it across the x-axis. This affects the steepness and direction of the graph's slopes.

Asymptotes of the 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 essential for correctly sketching the graph.