Graph y=-2sin(2x)

Problem

y=−2*sin(2*x)

Solution

1. Identify the amplitude. The amplitude is the absolute value of the coefficient of the sine function, |a|=|−2|=2 This means the graph oscillates between y=2 and y=−2

2. Determine the period. The period P of a sine function y=a*sin(b*x) is calculated using the formula P=(2*π)/b Here, b=2 so P=(2*π)/2=π

3. Find the phase shift and vertical shift. Since there are no horizontal or vertical translations added to the argument or the function, the phase shift is 0 and the vertical shift is 0

4. Identify key points. Divide the period into four equal intervals of length π/4 Starting from x=0 the key xvalues are 0,π/4,π/2,(3*π)/4,π

5. Calculate the y-coordinates. Evaluate the function at the key xvalues:

• At x=0 y=−2*sin(0)=0

• At x=π/4 y=−2*sin(π/2)=−2

• At x=π/2 y=−2*sin(π)=0

• At x=(3*π)/4 y=−2*sin((3*π)/2)=2

• At x=π y=−2*sin(2*π)=0

6. Sketch the graph. Plot the points (0,0) (π/4,−2) (π/2,0) ((3*π)/4,2) and (π,0) and connect them with a smooth sine wave. Note that the negative sign in front of the amplitude causes the graph to be reflected across the xaxis compared to a standard sine wave.

Final Answer

y=−2*sin(2*x)

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