Answer:
y = 7 csc(½ x) − 2
Explanation:
General form of a cosecant function is:
y = A csc(2π/T x + B) + C
where A is the amplitude, T is the period, B is the horizontal offset ("phase shift"), and C is the vertical offset ("midline").
The range is (-∞, -9] [5, ∞), so the midline is halfway between -9 and 5.
C = (-9 + 5) / 2
C = -2
The amplitude is half the difference between -9 and 5.
A = |-9 − 5| / 2
A = 7
The period is twice the distance between consecutive asymptotes.
T = 2 (2π − 0)
T = 4π
So far, we have:
y = 7 csc(½ x + B) − 2
We know there is an asymptote at x = 0. Cosecant is undefined at multiples of π, so:
½ (0) + B = kπ
B = kπ
B is any multiple of π. The simplest choice is B = 0.
y = 7 csc(½ x) − 2