Graph y=2sin(x)-1/2

Problem

y=2*sin(x)−1/2

Solution

1. Identify the parent function and its properties. The base function is y=sin(x) which has an amplitude of 1 a period of 2*π and a midline at y=0

2. Determine the amplitude. The coefficient of the sine function is 2 so the amplitude is |2|=2 This means the graph will oscillate 2 units above and below the midline.

3. Determine the vertical shift. The constant term −1/2 indicates a vertical shift downward. The new midline of the graph is the horizontal line y=−1/2

4. Calculate the range. Since the midline is at −1/2 and the amplitude is 2 the maximum value is −1/2+2=3/2 and the minimum value is −1/2−2=−5/2

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

• At x=0 y=2*sin(0)−1/2=−1/2

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

• At x=π y=2*sin(π)−1/2=−1/2

• At x=(3*π)/2 y=2*sin((3*π)/2)−1/2=2*(−1)−1/2=−5/2

• At x=2*π y=2*sin(2*π)−1/2=−1/2

6. Sketch the graph by plotting these key points and drawing a smooth sinusoidal wave through them, repeating the pattern every 2*π units.

Final Answer

y=2*sin(x)−1/2

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