Graph y=3cos(2x)

Problem

y=3*cos(2*x)

Solution

1. Identify the amplitude by looking at the coefficient of the cosine function. The amplitude A is the absolute value of the coefficient 3

A=|3|=3

2. Determine the period of the function using the coefficient of x which is b=2 The formula for the period P of a cosine function is P=(2*π)/b

P=(2*π)/2=π

3. Calculate the phase shift and vertical shift by observing the arguments. Since there is no constant added inside or outside the cosine 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/4 The xcoordinates are 0 π/4 π/2 (3*π)/4 and π

x=0⇒y=3*cos(0)=3

x=π/4⇒y=3*cos(π/2)=0

x=π/2⇒y=3*cos(π)=−3

x=(3*π)/4⇒y=3*cos((3*π)/2)=0

x=π⇒y=3*cos(2*π)=3

5. Plot the points and draw a smooth wave. The graph oscillates between y=3 and y=−3 with a full cycle completed every π units along the xaxis.

Final Answer

y=3*cos(2*x)

(The graph is a cosine wave with amplitude 3 period π and no phase shift.)

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