Graph y=2cos(x)-1

Problem

y=2*cos(x)−1

Solution

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

2. Determine the amplitude from the coefficient of the cosine term. Here, A=2 This means the graph will stretch vertically, reaching 2 units above and below the midline.

3. Identify the vertical shift from the constant term. The value d=−1 shifts the entire graph down by 1 unit. The new midline (average value) is the horizontal line y=−1

4. Calculate the range of the function. Since the midline is at −1 and the amplitude is 2 the maximum value is −1+2=1 and the minimum value is −1−2=−3

5. Determine the period and key points. The coefficient of x is 1 so the period remains 2*π Key points occur at x=0,π/2,π,(3*π)/2,2*π

6. Plot the key points over one period:

• At x=0 y=2*cos(0)−1=2*(1)−1=1

• At x=π/2 y=2*cos(π/2)−1=2*(0)−1=−1

• At x=π y=2*cos(π)−1=2*(−1)−1=−3

• At x=(3*π)/2 y=2*cos((3*π)/2)−1=2*(0)−1=−1

• At x=2*π y=2*cos(2*π)−1=2*(1)−1=1

7. Draw a smooth curve through these points to complete the graph.

Final Answer

To graph y=2*cos(x)−1 plot a cosine wave with amplitude 2 midline y=−1 and period 2*π oscillating between y=1 and y=−3

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