To solve this problem, we need to analyze the provided function type and the given specifications:
The function has the form:
[tex]\[ y = A \sin (B(x - C)) - D \][/tex]
We are given the following specific values:
- Amplitude [tex]\( A \)[/tex] is 2.
- There is an expansion factor [tex]\( B \)[/tex] of 2.
- The function shifts down [tex]\( D \)[/tex] by 3 units.
From the given information, we can determine the values of [tex]\( A \)[/tex], [tex]\( B \)[/tex], [tex]\( C \)[/tex], and [tex]\( D \)[/tex] as follows:
- Amplitude [tex]\( A \)[/tex]: The amplitude of a sine function is the coefficient that multiplies the sine term. Given that the amplitude is 2, we have:
[tex]\[ A = 2 \][/tex]
- Expansion [tex]\( B \)[/tex]: The expansion factor affects the period of the sine function. Given that there is an expansion of 2, we have:
[tex]\[ B = 2 \][/tex]
- Horizontal shift [tex]\( C \)[/tex]: The horizontal shift of the function as described by [tex]\( (x - C) \)[/tex] is not specified in the problem, which means there is no horizontal shift. Thus:
[tex]\[ C = 0 \][/tex]
- Vertical shift [tex]\( D \)[/tex]: The function shifts downward, represented by the term outside the sine function. Given that the shift down is 3 units, we have:
[tex]\[ D = 3 \][/tex]
Putting all these values together, we get:
[tex]\[ A = 2, \; B = 2, \; C = 0, \; D = 3 \][/tex]
Comparing this with the options provided:
a. [tex]\(\quad 0.5,3,0,2\)[/tex]
b. (Note: Option 'c' is not described)
c. [tex]\(\quad 0.5,0,2,3\)[/tex]
We see that none of the given options match the values derived:
[tex]\[ (2, 2, 0, 3) \][/tex]
Therefore, the correct values of [tex]\( A, B, C, \)[/tex] and [tex]\( D \)[/tex] for the given question are:
[tex]\[ A = 2, \; B = 2, \; C = 0, \; D = 3 \][/tex]
No correct option is provided among the given choices.
