Graph y=-4cos(2x)

Problem

y=−4*cos(2*x)

Solution

1. Identify the amplitude. The amplitude A is the absolute value of the coefficient of the cosine function.

|A|=|−4|=4

2. Determine the period. The period P of a cosine function y=a*cos(b*x) is calculated using the formula P=(2*π)/b

P=(2*π)/2=π

3. Find the phase shift and vertical shift. Since there are no horizontal or vertical translations added to the function, both shifts are zero.

Phase Shift=0

Vertical Shift=0

4. Identify key points. Divide the period into four equal intervals of π/4 to find the xcoordinates for the maximums, minimums, and intercepts.

x=0,π/4,π/2,(3*π)/4,π

5. Calculate the y-coordinates. Evaluate the function at the key xvalues. Note that the negative sign reflects the graph across the xaxis.

y(0)=−4*cos(0)=−4

y(π/4)=−4*cos(π/2)=0

y(π/2)=−4*cos(π)=4

y((3*π)/4)=−4*cos((3*π)/2)=0

y(π)=−4*cos(2*π)=−4

6. Sketch the graph. Plot the points (0,−4) (π/4,0) (π/2,4) ((3*π)/4,0) and (π,−4) and connect them with a smooth wave.

Final Answer

y=−4*cos(2*x)

The graph is a cosine wave with amplitude 4 period π and a reflection over the xaxis.

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