Graph y=-3cos(2x)

Problem

y=−3*cos(2*x)

Solution

1. Identify the parent function and the general form of the trigonometric equation. The equation follows the form y=A*cos(B*x) where A is the amplitude and B affects the period.

2. Determine the amplitude by taking the absolute value of the coefficient A Here, |A|=|−3|=3 The graph will oscillate between y=3 and y=−3

3. Identify the vertical reflection caused by the negative sign in front of the amplitude. Since A=−3 the graph of the cosine function is reflected across the xaxis, meaning it starts at a minimum point (0,−3) instead of a maximum.

4. Calculate the period using the formula P=(2*π)/B With B=2 the period is P=(2*π)/2=π One full cycle occurs over the interval [0,π]

5. Determine the key points by dividing the period into four equal increments. The increment size is π/4 The xcoordinates for the key points are 0 π/4 π/2 (3*π)/4 and π

6. Evaluate the function at these key points to find the ycoordinates:

• At x=0 y=−3*cos(0)=−3

• At x=π/4 y=−3*cos(π/2)=0

• At x=π/2 y=−3*cos(π)=3

• At x=(3*π)/4 y=−3*cos((3*π)/2)=0

• At x=π y=−3*cos(2*π)=−3

7. Plot the points and draw a smooth wave connecting (0,−3) (π/4,0) (π/2,3) ((3*π)/4,0) and (π,−3)

Final Answer

y=−3*cos(2*x)

(The graph is a cosine wave with amplitude 3 period π reflected over the xaxis, passing through (0,−3) (π/2,3) and (π,−3))

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