Final answer:
The period of the given function is π/2. The graph of y = tan 2 (x + π/2) has vertical asymptotes at x = kπ and passes through the points (k+1/2)π.
Step-by-step explanation:
The function given is y = tan 2 (x + π/2). To find the period of this function, we need to determine the period of the tangent function. The tangent function has a period of π radians, meaning it completes one full cycle from 0 to π and back to 0. The coefficient of 2 in front of the angle x indicates that the function completes two cycles within the period of the tangent function. Therefore, the period of the given function is π/2.
To graph the function, we can start by graphing the tangent function over one period. The tangent function has vertical asymptotes at x = π/2, 3π/2, 5π/2, etc., and it passes through x = 0 and x = π. With the given function, we need to shift the graph π/2 units to the left, which moves the vertical asymptotes to x = 0, π, 2π, etc., and the points x = -π/2 and x = 3π/2 now become the points where the function passes through. The graph will repeat this pattern every π/2 units.
Therefore, the graph of y = tan 2 (x + π/2) has a period of π/2 and repeats every π/2 units. It has vertical asymptotes at x = kπ, where k is an integer, and it passes through the points (k+1/2)π, where k is an integer.