Please help me understand how to Write the equation for each sine function

Question

Answer 1

for the 1st graph the period = 120 degrees

amplitude is 3

it's also been shifted left by 30 degrees

and shifted down by by 1

the midline is y=-1

take a standard sine function, f(x) = sinx, shift it left, shift down, increase the amplitude and decrease the period

x

to give it amplitude 3, multiply by 3 to get f(x) = 3sinx

shift down 1 by subtracting 1 to get f(x) =3sinx -1

now change the period and shift it left by 30 degrees

as is, the period = 360 or 2pi. to change to 120, the period is divided by 3. 120=360/3

f(x) = 3sin(3x) -1

shift the sine curve left by 30

f(x) =3sin(3(x+30)) -1

general equation is Asin(B(x+C)) +D

for f(x) = sinx, A=1, B=1, C= 0, D=0

A is the amplitude,

period = 2pi/B = 360/B

C is leftward horizontal shift

D is upward vertical shift

in the second graph

the amplitude remains the same, just 1, A=1

but there's a vertical shift up, by 3, D=3

a rightward shift by 40 degrees, C=-40

period = 180=360/B, B=2

f(x) = Asin(B(x+C))+D

f(x) = sin(2(x-40))+3