What is the graph of one cycle of a sine curve with the given characteristics? using the form y = a sin b theta, what is an equation for the sine curve

To start we need to determine its amplitude and the period.

Amplitude = 4
Period = 1/4 pi 
a < 0

The amplitude is the absolute value of a: that is absolute value of , and it is given that a<0 ; thus, if the amplitude is 4, but a is less than zero, you need to use -4 for a.

a= -4 

The period is usually 2pi, and in this case it is (1/4)* pi = pi/4

Everything has to go 8 times, so b = 8

This will give us a function:

y = -48sin (8*theta)

Below is the attached picture of the graph.

Letter c hope it