Graph y=cos(2x-pi/4)

Problem

y=cos(2*x−π/4)

Solution

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

2. Determine the amplitude |A| which is the coefficient of the cosine function. Here, A=1 so the amplitude is 1

3. Calculate the period using the formula P=(2*π)/B Since B=2 the period is (2*π)/2=π

4. Find the phase shift by factoring the expression inside the cosine: 2*x−π/4=2*(x−π/8) The phase shift C is π/8 to the right.

5. Determine the vertical shift D Since there is no constant added to the function, D=0 meaning the midline is the x-axis.

6. Identify key points for one cycle starting at the phase shift x=π/8 The interval for one cycle is [π/8,π/8+π] which is [π/8,(9*π)/8]

7. Divide the period into four equal increments of π/4 to find the x-coordinates of the maximums, minimums, and intercepts: π/8 (3*π)/8 (5*π)/8 (7*π)/8 and (9*π)/8

Final Answer

y=cos(2*x−π/4)

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