Graph y=2cos(1/3x)

Problem

y=2*cos(1/3*x)

Solution

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

A=|2|=2

2. Determine the period of the function using the formula P=(2*π)/b where b is the coefficient of x Here, b=1/3

P=(2*π)/(1/3)=6*π

3. Calculate the phase shift and vertical shift. Since there are no constants added inside or outside the cosine function, both shifts are zero.

Phase Shift=0

Vertical Shift=0

4. Find the key points for one full cycle by dividing the period into four equal intervals. The interval width is (6*π)/4=(3*π)/2

(x_0)=0⇒y=2*cos(0)=2

(x_1)=(3*π)/2⇒y=2*cos(π/2)=0

(x_2)=3*π⇒y=2*cos(π)=−2

(x_3)=(9*π)/2⇒y=2*cos((3*π)/2)=0

(x_4)=6*π⇒y=2*cos(2*π)=2

5. Plot the points and draw a smooth cosine wave. The graph oscillates between y=2 and y=−2 with a total horizontal length of 6*π for one cycle.

Final Answer

y=2*cos(1/3*x)

The graph is a cosine wave with amplitude 2 period 6*π starting at (0,2) passing through (3*π,−2) and returning to (6*π,2)

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