Graph y=4cos(x)-1

Problem

y=4*cos(x)−1

Solution

1. Identify the parent function and its properties. The base function is y=cos(x) which has an amplitude of 1 a period of 2*π and oscillates around the x-axis (y=0.

2. Determine the amplitude. The coefficient of the cosine function is 4 so the amplitude is |4|=4 This means the graph stretches vertically, reaching 4 units above and below the midline.

3. Determine the midline. The constant term −1 shifts the entire graph down by 1 unit. The new midline (vertical center) is the horizontal line y=−1

4. Calculate the range. Since the midline is at −1 and the amplitude is 4 the maximum value is −1+4=3 and the minimum value is −1−4=−5 The range is [−5,3]

5. Identify key points for one period [0,2*π]

• At x=0 y=4*cos(0)−1=4*(1)−1=3 (Maximum)

• At x=π/2 y=4*cos(π/2)−1=4*(0)−1=−1 (Midline)

• At x=π y=4*cos(π)−1=4*(−1)−1=−5 (Minimum)

• At x=(3*π)/2 y=4*cos((3*π)/2)−1=4*(0)−1=−1 (Midline)

• At x=2*π y=4*cos(2*π)−1=4*(1)−1=3 (Maximum)

6. Sketch the curve by plotting these key points and connecting them with a smooth, periodic wave.

Final Answer

y=4*cos(x)−1

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