Graph y=3cos(1/3x)

Problem

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

Solution

1. Identify the amplitude by looking at the coefficient in front of the cosine function. The amplitude A is |3| which means the graph oscillates between y=3 and y=−3

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

P=(2*π)/(1/3)

P=6*π

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

4. Calculate 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

• Start at x=0 y=3*cos(0)=3

• Next point x=(3*π)/2 y=3*cos(π/2)=0

• Midpoint x=3*π y=3*cos(π)=−3

• Next point x=(9*π)/2 y=3*cos((3*π)/2)=0

• End of cycle x=6*π y=3*cos(2*π)=3

5. Sketch the curve by plotting these five key points (0,3) ((3*π)/2,0) (3*π,−3) ((9*π)/2,0) and (6*π,3) and connecting them with a smooth wave.

Final Answer

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

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