What is the amplitude, period and phase shift for y = 1/4 cos (3x +7)? Thank you, please show work.
rewrite it in the more general form as
y = (1/4) cos 3(x + 7/3)
compared to y = a cos k(Ø + p)
the amplitude is 1/4
the period is 2π/3
the phase shift is 7/3 to the left
for your equation in the original form a common mistake would be to state that the sphase shift is 7.
Note the step I took to avoid that error.
To find the amplitude, period, and phase shift of the given function y = 1/4 cos (3x + 7), we can use the general form of the cosine function:
y = A cos (Bx - C)
In this equation:
- A represents the amplitude
- B represents the coefficient of x, which determines the period
- C represents the phase shift
Comparing the given function with the general form, we can identify the values for A, B, and C.
Amplitude (A):
The coefficient in front of the cosine function, 1/4, represents the amplitude. So, the amplitude is 1/4.
Period (P):
The period of the cosine function is given by the formula 2π/B. In the given function, B is 3. Therefore, the period is 2π/3.
Phase Shift (C):
The phase shift represents how much the graph is shifted horizontally. To find the phase shift, we need to solve the equation Bx - C = 0 for x. In the given function, B = 3 and C = 7. Let's solve for x:
3x + 7 = 0
3x = -7
x = -7/3
So, the phase shift is -7/3.
In summary:
- Amplitude (A) = 1/4
- Period (P) = 2π/3
- Phase Shift (C) = -7/3
Please note that the phase shift is negative, which means the graph is shifted to the right by 7/3 units.