1) what is the phase shift of f(x) = -2sin(3x-pi)+1
2) What is the period of f(x) = -2sin(3x-pi)+1
In relation to what function?
f(x) = -2sin(3x-pi)+1 = f(x) = -2 sin (3(x-π/3) ) + 1 <---- a more standard form
if you are comparing f(x) = -2sin(3x-pi)+1 with f(x) = -2sin(3x)
you would have translated f(x) = -2sin(3x) one unit up and π/3 to the right
for the period, if y = a sin kx, the period is 2π/k units
so for yours .....
3x-π = 3(x - π/3)
so the shift is π/3 to the right
the period of sin(kx) is 2π/k
To determine the phase shift of a function, we need to set the argument of the sine function equal to zero and solve for 'x'. For the function f(x) = -2sin(3x-pi)+1:
1) Phase shift of f(x) = -2sin(3x-pi)+1:
Setting the argument 3x - pi equal to zero:
3x - pi = 0
Solving for 'x':
3x = pi
x = pi/3 + n*(2pi/3), where n is an integer
Therefore, the phase shift is pi/3.
2) To find the period of a function, we need to determine the value of 'b' in the general form of the sine function, f(x) = a*sin(bx). In this case, the function is f(x) = -2sin(3x-pi)+1.
Comparing the given function to the general form, we can see that b = 3.
The period is calculated using the formula T = (2pi)/b:
T = (2pi)/3
Hence, the period of f(x) = -2sin(3x-pi)+1 is (2pi)/3.
To find the phase shift and the period of the given function f(x) = -2sin(3x-pi)+1, we can analyze the components of the function.
1) Phase Shift:
The general formula for a sinusoidal function is f(x) = A*sin(Bx-C)+D, where A represents the amplitude, B represents the frequency, C represents the phase shift, and D represents the vertical shift.
In our given function, f(x) = -2sin(3x-pi)+1, we can observe that the phase shift occurs inside the parentheses, which is (3x-pi). To find the phase shift, we set the argument inside the sin function equal to zero (0):
3x - pi = 0
3x = pi
x = pi/3
Therefore, the phase shift of the function f(x) = -2sin(3x-pi)+1 is pi/3.
2) Period:
The period of a sine or cosine function can be found using the formula:
T = 2π/B
In our given function, f(x) = -2sin(3x-pi)+1, the coefficient of x inside the sin function is 3, which represents the frequency. Therefore, the period (T) is given by:
T = 2π/3
So, the period of the function f(x) = -2sin(3x-pi)+1 is 2π/3.