Which cosine function has maximum of -2, and a period of 2pi/3
2 cos 3O
To find the cosine function with a maximum of -2 and a period of 2Ï€/3, we can start by considering the general equation for a cosine function:
y = A * cos(Bx + C) + D
where A represents the amplitude, B is the frequency, C is the phase shift, and D is the vertical shift.
Since we want the function to reach a maximum of -2, the amplitude (A) can be determined as A = |-2| = 2.
The period of the cosine function is given as 2Ï€/3. The formula for determining the frequency (B) based on the period is B = 2Ï€ / period. Substituting the given period, we have:
B = 2Ï€ / (2Ï€/3) = 3
Next, we need to find the phase shift (C). We know that the cosine function has a maximum at x = 0, which means that the phase shift is 0: C = 0.
Finally, we need to find the vertical shift (D). Since the function has a maximum of -2, this means that D = maximum value - amplitude = -2 - (-2) = -2 + 2 = 0.
Putting it all together, the cosine function with a maximum of -2 and a period of 2Ï€/3 is:
y = 2 * cos(3x)
To find the equation of a cosine function with a maximum value of -2 and a period of 2Ï€/3, we need to consider the general form of a cosine function:
f(x) = A * cos(Bx - C) + D
where:
- A represents the amplitude,
- B represents the frequency (or number of cycles in 2Ï€),
- C represents a phase shift, and
- D represents a vertical shift.
Given that the maximum value is -2, the amplitude (A) of the function is 2 since the cosine function oscillates between -A and +A. However, in this case, we want the maximum value to be -2, so we take the negative value.
Given that the period is 2Ï€/3, the frequency (B) is equal to 2Ï€ divided by the period, which is:
B = 2Ï€ / (2Ï€/3)
B = 3
The phase shift (C) is 0 because there is no horizontal shift.
Finally, since we want the maximum to be -2, the vertical shift (D) is -2.
Putting it all together, the equation of the cosine function is:
f(x) = -2 * cos(3x) - 2
Therefore, the cosine function with a maximum of -2 and a period of 2Ï€/3 is f(x) = -2 * cos(3x) - 2.
y = -3 + cos 2 pi (3/2pi) x
or
y = -3 + cos 3 x