Final answer:
The correct modification to have a period of 8Π is division of the x variable by 4 inside the sine function, making the correct option (d).
Step-by-step explanation:
To modify the function f(x) = sin(x) to have a period of 8Π, we would need to look at how the period of a sine function is determined. The general form of a sine function is f(x) = sin(Bx), where 2Π/|B| gives us the period of the sine wave. Since the period we desire is 8Π, we need to find the value of B that makes this true.
The original period is 2Π, so to make the period 8Π, which is four times longer, we need to divide the coefficient of x inside the sine function by 4. Therefore, the modified function will be f(x) = sin(x/4) or f(x) = sin(0.25x). This alteration to the function changes its period, without affecting its range or the fact that it oscillates between +1 and -1.
In summary:
- Original function: f(x) = sin(x), Period: 2Π
- Modified function: f(x) = sin(x/4), Period: 8Π
The correct modification to have a period of 8Π is division of the x variable by 4 inside the sine function, making the correct option (d).