Find Amplitude, Period, and Phase Shift y=6cos(pi-2x)+7

Problem

y=6*cos(π−2*x)+7

Solution

1. Rewrite the function into the standard form y=A*cos(B*(x−C))+D by rearranging the terms inside the cosine and factoring out the coefficient of x

y=6*cos(−2*x+π)+7

2. Factor out the value −2 from the expression inside the parentheses to isolate x

y=6*cos(−2*(x−π/2))+7

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

y=6*cos(2*(x−π/2))+7

4. Identify the parameters from the standard form y=A*cos(B*(x−C))+D where A=6 B=2 C=π/2 and D=7

Amplitude=|A|=|6|=6

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

Period=(2*π)/2=π

6. Determine the phase shift by identifying the value of C from the factored form.

Phase Shift=π/2

Final Answer

Amplitude=6,Period=π,Phase Shift=π/2

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