Question

4) 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

Answer:

a) 2π;  [-3,5]; b) 2π; [-2,2]

Step-by-step explanation:

(a) y = 1 + 4sin x

The general form of a sine function is  

y = A(sin(B(x - h)) + k

where

   |A| = amplitude

2π/B = T =period

     h = phase shift (horizontal shift, to the right if x > 0)

     k = vertical shift (midline is y = k)

Your function is

y = 1 + 4sin(x)  

Comparing terms, we find that

h = 0

k = 1

A = 4

B = 1

The amplitude is 4.

The midline is y = 1.

The horizontal shift is 0.

(i) Period

T = 2π/B = 2π/1 = 2π

(ii) Graph

The graph of the function is the first figure below.

(iii) Image set

The image of a function is the set of all possible values it can have.

If f(x)= 1 + 4sin(x), the domain of the function is (-∞,∞).

The corresponding values of y can take any value from -3 to 5.

The image set is [-3,5].

(b) y = 2sin(x - 3)

h = 3; k = 0; A = 2; B = 1

The amplitude is 2

The midline is y = 0.

The horizontal shift is 3 radians to the right.

(i) Period

T = 2π/B = 2π/1 = 2π

(ii) Graph

The graph of the function is the second figure below.

(iii)Image set

If f(x) = 2sin(x - 3), the domain of the function is (-∞,∞)

The corresponding values of y can take any value from -2 to 2.

The image set is [-2,2].