Graph y=1/2*cos(4x)

Problem

y=1/2*cos(4*x)

Solution

1. Identify the amplitude by looking at the coefficient of the cosine function. The amplitude A is the absolute value of the leading coefficient.

A=|1/2|=1/2

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

P=(2*π)/4=π/2

3. Calculate the phase shift and vertical shift by examining the arguments and constants. Since there is no horizontal or vertical translation added to the function, both shifts are zero.

Phase Shift=0

Vertical Shift=0

4. Find the key points for one cycle by dividing the period into four equal intervals. The interval width is π/2÷4=π/8 The xcoordinates are 0 π/8 π/4 (3*π)/8 and π/2

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

x=0⇒y=1/2*cos(0)=1/2

x=π/8⇒y=1/2*cos(π/2)=0

x=π/4⇒y=1/2*cos(π)=−1/2

x=(3*π)/8⇒y=1/2*cos((3*π)/2)=0

x=π/2⇒y=1/2*cos(2*π)=1/2

6. Sketch the graph by plotting these points (0,1/2) (π/8,0) (π/4,−1/2) ((3*π)/8,0) and (π/2,1/2) and connecting them with a smooth wave.

Final Answer

y=1/2*cos(4*x)

The graph is a cosine wave with amplitude 1/2 period π/2 and no shifts.

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