Graph y=cos(2x-pi)

Problem

y=cos(2*x−π)

Solution

1. Identify the parent function and the general form of the trigonometric equation. The equation is in the form y=A*cos(B*(x−C))+D

2. Determine the amplitude by looking at the coefficient A in front of the cosine function. Here, A=1 so the amplitude is 1

3. Calculate the period using the formula P=(2*π)/B In this equation, B=2

P=(2*π)/2=π

4. Find the phase shift by setting the argument of the cosine function to zero and solving for x

2*x−π=0

⇒2*x=π

⇒x=π/2

The phase shift is π/2 to the right.

5. Determine the vertical shift by looking at the constant D Here, D=0 so the midline is the x-axis (y=0.

6. Identify key points for one cycle starting at the phase shift x=π/2 and ending at x=π/2+π=(3*π)/2 Divide the period into four equal intervals of π/4

• Start: (π/2,1)

• Quarter: ((3*π)/4,0)

• Midpoint: (π,−1)

• Three-quarters: ((5*π)/4,0)

• End: ((3*π)/2,1)

Final Answer

y=cos(2*x−π)

The graph is a cosine wave with amplitude 1 period π phase shift π/2 and midline y=0

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