Vertical translations?

Question

A sine function has an amplitude of 3, period of 6pi, a horizontal shift of 3pi/2, and vertical shift of -1. What is the y-value of the positive function when x=2pi?

Answer 1

Sin function 

 

y= a sin(bx - c ) + d

 

where 

a = amplitude

 

period = 2pi/b

 

horizontal shift =-c/b

 

vertical shift = d

 

Here given that

 

a = amplitude =3

period = 2pi/b

 

6pi = 2pi/b

 

b=2pi/6pi

 

b=1/3

horizontal shift =-c/b

 

3pi/2 =-c/ 1/3

 

 

3pi/2 * 1/3 =-c

 

pi/2=-c

 

c=-pi/2

vertical shift = d

 

-1 =d

 

d=-1

 

substitute all value in our sin function equation

 

y= a sin(bx - c ) + d

 

y = 3 sin((1/3) x + pi/2 ) - 1

 

given that x= 2pi

 

y= 3 sin(1/3 * 2pi +pi/2)-1

 

   = 3 sin (2pi/3 +pi/2)-1

   

   = 3 sin(4pi/6 +3pi/6)-1

 

  y = 3 sin(7pi/6) -1