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2 answers:
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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].
