Question 212798
write an equation of the sine function with amplitude 3,period 3pi/2, and phase shift pi/4

a)y=3sin(3x/2 - pi/4)
b)y=-3sin(4x/3 - pi/4)
c)y=-3sin(3x/2 - 3pi/8)
d)y=3sin(4x/3 - pi/3)

<pre><font size = 4 color = "indigo"><b> 
{{{y = Asin(Bx-C)}}}, with B and C positive.

has amplitude {{{abs(A)}}}, period {{{2pi/B}}}, and phase shift {{{C/B}}},
The shift is right if {{{C}}} is positive and left if {{{C}}} 
is negative.

We'll test each possible answer:

a){{{y=3sin(3x/2 - pi/4)}}}

has amplitude {{{abs(A)=abs(3)=3}}}, 
period {{{ 2pi/B=(2pi)/(3/2)=(2pi)(2/3) = 4pi/3 }}}
and phase shift {{{C/B=(pi/4)/(3/2)=(pi/4)(2/3)=2pi/12=pi/6}}}.

b){{{y=-3sin(4x/3 - pi/4)}}}

has amplitude {{{abs(A)=abs(-3)=3}}}, 
period {{{2pi/B=(2pi)/(4/3)=(2pi)(3/4) = 6pi/4=3pi/2}}}
and phase shift {{{C/B=(pi/4)/(4/3)=(pi/4)(3/4)=3pi/16}}}.

c){{{y=-3sin(3x/2 - 3pi/8)}}}

has amplitude {{{abs(A)=abs(3)=3}}}, 
period {{{2pi/B=(2pi)/(3/2)=(2pi)(2/3) = 4pi/3}}}
and phase shift {{{C/B=(3pi/8)/(3/2)=(3pi/8)(2/3)=6pi/24=pi/4}}}.

d){{{y=3sin(4x/3 - pi/3)}}}

has amplitude {{{abs(A)=abs(3)=3}}}, 
period {{{2pi/B=(2pi)/(4/3)=(2pi)(3/4) = 6pi/4=3pi/2}}}
and phase shift {{{C/B=(pi/3)/(4/3)=(pi/3)(3/4)=3pi/12=pi/4}}}

So (d) is the only correct choice.

Edwin</pre>