Question

Determine o período e o conjunto imagem, construindo o gráfico de um período completo para cada função dada.a) por f(x) = 1+ 4 sen x.b) por f(x) = 2sen (x-3)

Answer 1

Explanation:

The general form of a sinusoidal function is given by:

`f(x)=Asin(\omega x)` (1)

where A is the amplitude, and the period is given by:

`T=(2\pi)/(\omega)`

The image of a function is the available values of the function f(x) in the domain of f(x). For sinusoidal function the image of f(x) is determined with the aid of the amplitude. For example, in the case of (1), f(x) varies from -A to A.

Due to the information given above you have (the plots are attached below):

a)

`f(x)=1+4sin(x)\\\\T=(2\pi)/(1)=2\pi\\\\Im \ f(x)= [1-4,1+4]=[-3,5]`

b)

`f(x)=2sin(x-3)\\\\T=(2\pi)/(1)=2\pi\\\\Im\ f(x)=[2,-2]`

The -3 in the argument of the sin function does not modify the period of the function, that number is only a phase shift of the graph (see the plot)