How to find the amplitude, period, phase shift, vertical shift, and the equation of the primary function of the function:
y = -cot(1/3x - pi/6)
how to find where the period begins and ends
To find the amplitude, period, phase shift, and vertical shift of the function y = -cot(1/3x - pi/6), you can use the general form of the trigonometric function:
y = A * f(B(x - C)) + D
where:
A is the amplitude,
B determines the period,
C determines the phase shift,
D determines the vertical shift, and
f is the primary trigonometric function (in this case, cot).
1. Amplitude (A):
Since the function is in the form -cot, the maximum and minimum values of the cotangent function are ±∞. Therefore, the amplitude does not exist for this function.
2. Period (P):
The period of the cotangent function is π/|B|. In this case, B = 1/3. So, the period is π/(1/3) = 3π.
3. Phase Shift (C):
To find the phase shift, set the argument of the cotangent function (1/3x - pi/6) equal to zero and solve for x:
1/3x - pi/6 = 0
1/3x = pi/6
x = (pi/6)/(1/3) = pi/2
The phase shift is pi/2 to the right, which means the function is horizontally shifted by pi/2.
4. Vertical Shift (D):
Since there is no coefficient added to the cotangent function, the vertical shift (D) is 0.
5. Equation of the primary function:
Since the primary trigonometric function is cot, the equation of the primary function is simply cot(x).
Now, to find where the period begins and ends, you can use the phase shift and period information:
The period begins at x = -P/2 + C and ends at x = P/2 + C.
In this case, the period starts at x = -3π/2 + pi/2 = -π and ends at x = 3π/2 + pi/2 = 2π. Therefore, the period of this function starts at -π and ends at 2π.