Answer:
The given functions are
[tex]f(x)=1+4sinx[/tex]
[tex]f(x)=2sin(x-3)[/tex]
The sine function has a standard period of [tex]2 \pi[/tex] by definition. However, this might change if we use a factor as coefficient of the x-varible, but in this case we don't have that.
Therefore, the period of both trigonometric functions is [tex]2 \pi[/tex].
Now, the images of each function is the y-variable set values that defines each function.
So, the function [tex]f(x)=1+4sinx[/tex] has an image defined by the set [tex][-3,5][/tex]. It's impotant to notice that the range of a standard function is [-1,1], however, in this case, the function was shifted 1 unit up and it was streched by a factor of 4, that's why the standard image changes to [tex][-3,5][/tex].
About the second function [tex]f(x)=2sin(x-3)[/tex], the image set is [tex][-2,2][/tex], because the function was streched by a factor of 2.
Additionally, the image attached shows the graph of the given functions.
