Final answer:
The equation that represents a sine curve with an amplitude of 2, a period of π/4, and passes through the point (1/6, 0) is Option A: y = 2sin(8πx).
Step-by-step explanation:
The question is looking for an equation that represents a sine curve with specific characteristics: an amplitude of 2, a period of π/4, and it must pass through the point (1/6, 0). To find the period of the sine function, we use the formula Period = 2π / B where B is the coefficient of x inside the sine function. Therefore, for a period of π/4, we must have B = 2π / (π/4) = 8. Thus, the coefficient of x must be 8π to get the required period. With an amplitude of 2, the function begins with 2sin. Lastly, it must pass through (1/6, 0), which means when x = 1/6, the sine function should equal 0. Plugging x into the equation, we get sin(8π * 1/6) = sin(4π/3), which is 0, satisfying the condition.
Hence, the appropriate equation is Option A: y = 2sin(8πx).