Answer:
[tex]f(x)=\frac{3}{4}\sin (2x)+3[/tex]
Step-by-step explanation:
We have the function, [tex]f(x)=\sin x[/tex]
It is required to form a function with period [tex]\pi[/tex], shifted vertically 3 units upwards and having amplitude = [tex]\frac{3}{4}[/tex]
Now, as we know, 'If a function f(x) has the period P, then f(bx) will have period [tex]\frac{P}{|b|}[/tex]'.
Since, the new function need to have period [tex]\pi[/tex], that is the value [tex]\frac{P}{|b|}=\pi[/tex] i.e. [tex]\frac{2\pi}{|b|}=\pi[/tex].
So, b= 2 implies the new function is [tex]f(x)=\sin (2x)[/tex]
Further, as the function need to be vertically shifted 3 units upwards, we get the new function, [tex]f(x)=\sin (2x)+3[/tex].
Finally, the amplitude of the function must be [tex]\frac{3}{4}[/tex], this means that the maximum and minimum value of the function is [tex]\frac{3}{4}[/tex] and [tex]\frac{-3}{4}[/tex].
This gives us the transformed final function is [tex]f(x)=\frac{3}{4}\sin (2x)+3[/tex].
