Graph y=-2cos(x)

Problem

y=−2*cos(x)

Solution

1. Identify the parent function and its key characteristics. The parent function is y=cos(x) which has an amplitude of 1 a period of 2*π and starts at a maximum point (0,1)

2. Determine the amplitude of the given function. The coefficient in front of the cosine function is −2 The amplitude is the absolute value |−2|=2

3. Identify the vertical reflection caused by the negative sign. Since the coefficient is negative, the graph of y=cos(x) is reflected across the xaxis. Instead of starting at a maximum, the graph starts at a minimum point (0,−2)

4. Determine the period of the function. The coefficient of x is 1 so the period remains 2*π

5. Find the key points over one period [0,2*π] by dividing the period into four equal intervals.

• At x=0 y=−2*cos(0)=−2 (Minimum)

• At x=π/2 y=−2*cos(π/2)=0 (Intercept)

• At x=π y=−2*cos(π)=2 (Maximum)

• At x=(3*π)/2 y=−2*cos((3*π)/2)=0 (Intercept)

• At x=2*π y=−2*cos(2*π)=−2 (Minimum)

6. Sketch the curve by plotting these points and connecting them with a smooth, wave-like shape.

Final Answer

y=−2*cos(x)

The graph is a cosine wave with amplitude 2 period 2*π and a vertical reflection, passing through key points (0,−2) (π/2,0) (π,2) ((3*π)/2,0) and (2*π,−2)

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