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6. If a function is of the type [tex]y = A \sin (B(x - C)) - D[/tex] and the amplitude is 2, there is an expansion of 2, and a shift down of 3, what are the values of [tex]A, B, C,[/tex] and [tex]D[/tex], respectively?

A. [tex]0.5, 3, 0, 2[/tex]

B. [tex]2, 0.5, 3, 0[/tex]

C. [tex]0.5, 0, 2, 3[/tex]

D. [tex]2, 0.5, 0, 3[/tex]



Answer :

GinnyAnswer
To solve the problem, we need to determine the values of [tex]\( A \)[/tex], [tex]\( B \)[/tex], [tex]\( C \)[/tex], and [tex]\( D \)[/tex] in the function [tex]\( y = A \sin (B(x - C)) - D \)[/tex] given the following conditions:

1. The amplitude is 2.
2. There is an expansion of 2.
3. There is a shift down of 3.

Let's break down these conditions step-by-step:

1. Amplitude (A):
The amplitude of a trigonometric function [tex]\( y = A \sin(\dots) \)[/tex] is the coefficient [tex]\( A \)[/tex] in front of the sine function. Given that the amplitude is 2:
[tex]\[ A = 2 \][/tex]

2. Expansion (B):
The expansion factor relates to the coefficient inside the argument of the sine function. When we say there is an expansion of 2, it means the period of the function is multiplied by 2. In other words, [tex]\( B \)[/tex] is the reciprocal of the expansion factor. Given an expansion of 2:
[tex]\[ B = \frac{1}{2} = 0.5 \][/tex]

3. Horizontal Shift (C):
The horizontal shift affects [tex]\( C \)[/tex] in the expression [tex]\( \sin (B(x - C)) \)[/tex]. No horizontal shift is mentioned in the problem, so [tex]\( C \)[/tex] must be:
[tex]\[ C = 0 \][/tex]

4. Vertical Shift (D):
The function is shifted vertically downward by 3 units. In our function [tex]\( y = A \sin(B(x - C)) - D \)[/tex], this downward shift means [tex]\( D \)[/tex] is positive 3:
[tex]\[ D = 3 \][/tex]

Putting it all together, the values are:
[tex]\[ A = 2, \quad B = 0.5, \quad C = 0, \quad D = 3 \][/tex]

Therefore, the correct option is (d):
[tex]\[ \boxed{2, 0.5, 0, 3} \][/tex]

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