Find Amplitude, Period, and Phase Shift y=-4cos(3-2x)

Problem

y=−4*cos(3−2*x)

Solution

1. Rewrite the expression inside the cosine function to match the standard form y=A*cos(B*(x−C))+D

y=−4*cos(−2*x+3)

2. Factor out the coefficient of x inside the parentheses to clearly identify the horizontal components.

y=−4*cos(−2*(x−3/2))

3. Apply the even property of the cosine function, which states cos(−θ)=cos(θ) to simplify the expression.

y=−4*cos(2*(x−3/2))

4. Identify the parameters from the standard form y=a*cos(b*(x−h)) where a=−4 b=2 and h=3/2

5. Calculate the amplitude using the absolute value of the leading coefficient.

Amplitude=|−4|=4

6. Calculate the period using the formula (2*π)/|b|

Period=(2*π)/2=π

7. Determine the phase shift from the value of h

Phase Shift=3/2

Final Answer

Amplitude=4,Period=π,Phase Shift=3/2

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