Finding f(x) for Sin X

[learningmathh]

Hello everyone, I need to find f(x) for this function



How do I go about to find f(x) when sin x=

a*sin(k(x-c))+ d

 

[soroban]

Hello, learningmathh!

I need to find f(x) for this function.
View attachment 4625
How do I go about to find f(x) when sin x= a*sin(k(x-c)) + d

The amplitude is 2: \(\displaystyle a = 2\)
The period is \(\displaystyle \pi:\;k =2\)
There is no horizontal shift: \(\displaystyle c = 0\)
There is no vertical shift: \(\displaystyle d = 0\)

Therefore: \(\displaystyle f(x) \:=\:2\sin(2x)\)

 

[HallsofIvy]

The basic function, f(x)= sin(x) has a minimum value of -1 and a maximum value of 1. It has period \(\displaystyle 2\pi\).
Your graph shows a minimum of -2 and maximum of 2. Since the "middle value" is 0 just like sin(x) itself, there is no "d" to be added on. Your A= 2 so that -1 is expanded to -2 and 1 is expanded to 2. It has period \(\displaystyle \pi\) so you want k(x- c) to go from 0 to \(\displaystyle 2\pi\) as x goes from 0 to \(\displaystyle \pi\). That is, you want \(\displaystyle k(0- c)= -kc= 0\) and \(\displaystyle k(\pi- c)= 2\pi\).
From the first, \(\displaystyle -kc= 0\), either k= 0 or c= 0. If k= 0 then \(\displaystyle k(\pi- c)= 0(\pi- c)= 0\), not \(\displaystyle 2\pi\). So we must have c= 0 and then \(\displaystyle k(\pi- 0)= k\pi= 2\pi\) and \(\displaystyle k= 2\).

 

[learningmathh]

Thanks very much understood it noe?