Given the function y = 3cos(3x + pi), identify:
Amplitude (if applicable, give the answer in fractional form)
Period (in radians as a multiple of pi - note: do not write "rad" or "radians" in your answer)
Phase Shift (if the shift is right, enter + and if to the left, enter - before the number; if none, enter the number zero)
y = 3cos(3x + pi)
y = 3cos 3(x + π/3)
amplitude : 3
period = 2π/3
phase shift: π/3 to the left
find the domain and the rang for the following function
Y=3cos 3x
To determine the amplitude, period, and phase shift of the given function y = 3cos(3x + π), we can analyze the equation.
1. Amplitude:
The amplitude of a cosine function is the absolute value of the coefficient in front of the cosine function. In this case, the coefficient is 3. Therefore, the amplitude of the function is 3 (without any fractional form as the value is already given as a whole number).
2. Period:
The period of a cosine function can be determined by the coefficient of x inside the cosine function. In this case, the coefficient is 3x. To find the period, we divide 2π (a full cycle) by the coefficient, which gives us 2π/3. Hence, the period of the function is 2π/3 radians.
3. Phase Shift:
The phase shift of a cosine function is determined by the constant value inside the parentheses of the cosine function. In this case, the constant value is π. To find the phase shift, we need to set the expression inside the parentheses equal to zero and solve for x. Hence, 3x + π = 0, and by subtracting π from both sides, we get 3x = -π. Dividing both sides by 3, we find x = -π/3. Therefore, the phase shift is -π/3.
In summary:
Amplitude = 3
Period = 2π/3 radians
Phase Shift = -π/3