State the amplitude, period, phase shift, and vertical shift of
f(x)=2cos(1/2x-pi/6)-1/3
To determine the amplitude, period, phase shift, and vertical shift of the function f(x) = 2cos(1/2x - π/6) - 1/3, we can follow these steps:
1. Amplitude: The amplitude measures the maximum displacement from the midline of the graph. For a cosine function, the amplitude is the coefficient of the cosine term. In this case, the amplitude is 2.
2. Period: The period measures the length of one complete cycle of the graph. To determine the period for a cosine function, we use the formula: Period = 2π / abs(b), where b is the coefficient of x. In this case, b = 1/2. Therefore, the period is 2π / abs(1/2) = 4π.
3. Phase Shift: The phase shift determines the horizontal translation of the graph. To find the phase shift, we set the expression inside the cosine function, 1/2x - π/6, equal to zero and solve for x. In this case, we have 1/2x - π/6 = 0. By adding π/6 to both sides and multiplying by 2, we get x = π/3. The phase shift is π/3.
4. Vertical Shift: The vertical shift represents the upward or downward shift of the graph. In this case, the vertical shift is -1/3, indicating a downward shift of 1/3 unit.
In summary:
Amplitude: 2
Period: 4π
Phase Shift: π/3
Vertical Shift: -1/3