Find the period (in radians), the phase shift (in radians) and the vertical shift of the graph of the function below.
y= 3 sec (ѳ/2- 3π/4)
The general form of the secant function is y = Asec(B(x - C)) + D, where A is the amplitude, B is the period, C is the phase shift, and D is the vertical shift.
In this case, we have:
A = 3 (amplitude)
B = 2 (period = 2π/B)
C = 3π/4 (phase shift)
D = 0 (vertical shift)
Therefore, the period is 2π/B = 2π/2 = π radians.
The phase shift is C = 3π/4 radians.
The vertical shift is D = 0.
wrong the answer is period: 4π
phase shift: Right 3π2
vertical Shift: None
show me the steps to get here
Apologies for the mistake. Here are the correct steps to find the period, phase shift, and vertical shift of the function y = 3sec(Ѳ/2 - 3π/4):
1. Period:
The period of the secant function is given by the formula: period = 2π/B
In this case, B = 1/2 (Since it is in the form Ѳ/2).
So, period = 2π / (1/2) = 4π radians.
2. Phase Shift:
To find the phase shift, we need to set the argument of the secant function equal to 0 and solve for Ѳ:
(Ѳ/2 - 3π/4) = 0
Ѳ/2 = 3π/4
Ѳ = 3π/2
This tells us the graph has been shifted to the right by 3π/2 radians.
3. Vertical Shift:
The function y = 3sec(Ѳ/2 - 3π/4) does not have a vertical shift since there is no constant added or subtracted to the secant function.
Therefore, the correct answers are:
Period: 4π
Phase Shift: 3π/2
Vertical Shift: None