Graph y=2cos(4x)

Problem

y=2*cos(4*x)

Solution

1. Identify the amplitude by looking at the coefficient in front of the cosine function. The amplitude is |A|=|2|=2 This means the graph oscillates between y=2 and y=−2

2. Determine the period using the formula P=(2*π)/b where b is the coefficient of x Here, b=4

P=(2*π)/4

P=π/2

3. Find the phase shift and vertical shift by examining the arguments. Since there is no constant added inside the cosine or outside the function, the phase shift is 0 and the vertical shift is 0

4. Calculate the key points for one cycle by dividing the period into four equal intervals of P/4=π/8 The xcoordinates are 0,π/8,π/4,(3*π)/8,π/2

5. Evaluate the function at these xcoordinates to find the yvalues:

• At x=0 y=2*cos(0)=2

• At x=π/8 y=2*cos(π/2)=0

• At x=π/4 y=2*cos(π)=−2

• At x=(3*π)/8 y=2*cos((3*π)/2)=0

• At x=π/2 y=2*cos(2*π)=2

6. Plot the points and draw a smooth cosine wave passing through (0,2) (π/8,0) (π/4,−2) ((3*π)/8,0) and (π/2,2)

Final Answer

y=2*cos(4*x)* has amplitude *2* and period π/2

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