Find Amplitude, Period, and Phase Shift f(x)=3cos(1/4*(pix)+pi)-2

Problem

ƒ(x)=3*cos(1/4*(π*x)+π)−2

Solution

1. Identify the standard form of the trigonometric function ƒ(x)=A*cos(B*(x−C))+D or ƒ(x)=A*cos(B*x+C)+D

2. Determine the amplitude by taking the absolute value of the coefficient A

|A|=|3|

Amplitude=3

3. Identify the coefficient B from the expression inside the cosine, which is π/4*x+π

B=π/4

4. Calculate the period using the formula P=(2*π)/|B|

P=(2*π)/π/4

P=2*π⋅4/π

Period=8

5. Find the phase shift by setting the argument of the cosine to zero and solving for x or using the formula Phase Shift=−C/B

π/4*x+π=0

π/4*x=−π

x=−4

Final Answer

Amplitude=3,Period=8,Phase Shift=−4

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