Questions: Sketch two periods of the graph for the following function. Identify the stretching factor, period, and asymptotes. f(x) = tan(π/2 x)

Solution Steps

The given function is \( f(x) = \tan\left(\frac{\pi}{2} x\right) \). The stretching factor is determined by the coefficient of \( x \) inside the tangent function. Here, the coefficient is \( \frac{\pi}{2} \).

The period of the standard tangent function \( \tan(x) \) is \( \pi \). For the function \( \tan\left(\frac{\pi}{2} x\right) \), the period is given by: \[ \text{Period} = \frac{\pi}{\frac{\pi}{2}} = 2 \]

The asymptotes of the standard tangent function \( \tan(x) \) occur at \( x = \frac{\pi}{2} + k\pi \) for any integer \( k \). For the function \( \tan\left(\frac{\pi}{2} x\right) \), the asymptotes occur at: \[ \frac{\pi}{2} x = \frac{\pi}{2} + k\pi \implies x = 1 + 2k \quad \text{for any integer } k \]

Final Answer