Graph y=2cos(2x)

Problem

y=2*cos(2*x)

Solution

1. Identify the parent function and the general form of the trigonometric equation. The given equation is in the form y=A*cos(B*x) where A is the amplitude and B is used to find the period.

2. Determine the amplitude, which is the absolute value of the coefficient A

|A|=|2|=2

This means the graph oscillates between y=2 and y=−2

3. Calculate the period of the function using the formula P=(2*π)/B

P=(2*π)/2=π

The graph completes one full cycle over an interval of π

4. Find the phase shift and vertical shift. Since there are no horizontal or vertical translations added to the function, both the phase shift and vertical shift are 0

5. Determine the key points for one period by dividing the period into four equal intervals. The interval width is π/4

• Start at x=0 y=2*cos(0)=2

• At x=π/4 y=2*cos(π/2)=0

• At x=π/2 y=2*cos(π)=−2

• At x=(3*π)/4 y=2*cos((3*π)/2)=0

• At x=π y=2*cos(2*π)=2

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

Final Answer

y=2*cos(2*x)

The graph is a cosine wave with amplitude 2 period π and key points at (0,2) (π/4,0) (π/2,−2) ((3*π)/4,0) and (π,2)

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