Graph y=cos(6x)

Problem

y=cos(6*x)

Solution

1. Identify the parent function and its properties. The function is of the form y=A*cos(B*x) where the parent function is y=cos(x)

2. Determine the amplitude, which is the absolute value of the coefficient A Here, A=1 so the amplitude is 1 This means the graph oscillates between y=−1 and y=1

3. Calculate the period using the formula P=(2*π)/|B| In this function, B=6

P=(2*π)/6

P=π/3

4. Find the key points by dividing the period into four equal intervals. The increment is P/4=(π/3)/4=π/12 The xcoordinates for one cycle starting at x=0 are 0 π/12 π/6 π/4 and π/3

5. Evaluate the function at these key xvalues to find the ycoordinates.

• At x=0 y=cos(0)=1

• At x=π/12 y=cos(π/2)=0

• At x=π/6 y=cos(π)=−1

• At x=π/4 y=cos((3*π)/2)=0

• At x=π/3 y=cos(2*π)=1

6. Sketch the curve by plotting these points (0,1) (π/12,0) (π/6,−1) (π/4,0) and (π/3,1) and connecting them with a smooth wave.

Final Answer

y=cos(6*x)* has amplitude *1* and period π/3

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