Question

Determine o período, a imagem e construa o gráfico de cada uma das funções abaixo: a) f(x) = 3sen(x) b) f(x) = 1 - sen(3x) c) f(x) = -1+2sen(0,5x)

Answer 1

Please read the answer below

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

b)

`f(x)=1-sin(3x)\\\\T=(2\pi)/(3)\\\\Im\ f(x)=[0 \ ,2]`

in the last case the first term of f(x) modifies the image of f(x) because the sinusoidal function is displaced to y=1.

c)

`f(x)=-1+2sin(0.5x)\\\\T=(2\pi)/(0.5)=4\pi\\\\Im\ f(x)=[-3,1]`