the function graphed to the right is of the form y = a sec bx + c or y = a csc bx + c for some a ≠ 0, b > 0…

Explanation:

Step1: Identify the type of function

The graph has vertical asymptotes at (x = \frac{\pi}{2}+k\pi,k\in\mathbb{Z}), so the function is of the form (y = a\csc(bx)+c).

Step2: Find the period

The period (T) of (y = a\csc(bx)+c) is given by (T=\frac{2\pi}{b}). From the graph, the period (T=\pi). Since (T = \frac{2\pi}{b}=\pi), we solve for (b): [b=\frac{2\pi}{T}=\frac{2\pi}{\pi}=2]

Step3: Find the vertical - shift (c)

The mid - line of the graph is (y = - 6), so (c=-6).

Step4: Find the amplitude (a)

The distance from the mid - line to a maximum or minimum is (|a|). The maximum value is (4) and the mid - line is (y=-6), so (|a| = 4 - (-6)=10). Since the graph opens downwards near the vertical asymptotes, (a=- 10).

Answer:

(y=-10\csc(2x)-6)