what is the equation involving sine that models a periodic function with an amplitude of 2, a period of 2ð/3, and a horizontal shift of 3.5 units?
To find the equation involving sine that models a periodic function with the given properties, we can start with the general form of a sine function:
y = A*sin(B(x - C)) + D
where A represents the amplitude, B is the frequency, C is the horizontal shift, and D is the vertical shift.
In this case, the given parameters are:
Amplitude (A) = 2
Period (T) = 2ð/3 (since the period T is related to the frequency B as T = 2ð/B)
Horizontal Shift (C) = 3.5
1. Amplitude (A):
The amplitude represents the maximum value that the function reaches. In this case, the amplitude is given as 2.
2. Frequency (B):
The frequency determines how many cycles (complete periods) the function completes within a given interval. The frequency can be calculated from the period using the formula B = 2ð/T. Given the period T of 2ð/3, the frequency B would be B = 2ð/(2ð/3) = 3.
3. Horizontal Shift (C):
The horizontal shift represents a translation of the graph to the left or right along the x-axis. In this case, the horizontal shift is given as 3.5 units.
Putting it all together, we have:
y = 2*sin(3(x - 3.5)) + D
Note: We haven't yet determined the vertical shift D, as it hasn't been given in the question. If a specific vertical shift is provided, you can substitute the value into the equation. Otherwise, you can leave it as + D, representing a general vertical shift.