Which cosine function has maximum of 0.5, a minimum of -0.5, and a period of 2pi/3?
To find the cosine function that satisfies the given conditions, we can use the general form of the cosine function:
f(x) = A * cos(B*(x - C)) + D
where A represents the amplitude, B is the frequency, C represents horizontal translation, and D represents vertical translation.
Given that the maximum value is 0.5, we can set A = 0.5. Similarly, since the minimum value is -0.5, D = -0.5.
Next, we need to determine the frequency. The period of the function is given as 2π/3. The formula for the period of a cosine function is T = 2π/B. Rearranging the equation, we find that B = 2π / T. Substituting T = 2π/3, we can calculate B as follows:
B = 2π / (2π/3)
B = 3
So, we have A = 0.5, B = 3, and D = -0.5. We are left to find the horizontal translation, C.
To determine C, we need to find the phase shift. The phase shift is given by the formula C = -k * (π / B), where k is an integer. Since the phase shift is not mentioned in the question, we assume it to be zero. Therefore, C = 0.
Combining all the values, we have:
f(x) = 0.5 * cos(3*(x - 0)) - 0.5
Simplifying further, we get:
f(x) = 0.5 * cos(3x) - 0.5
Thus, the cosine function that satisfies the given conditions is f(x) = 0.5 * cos(3x) - 0.5.