Graph y=3cos(x)-2

Problem

y=3*cos(x)−2

Solution

1. Identify the parent function and its basic properties. The parent 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 from the coefficient of the cosine term. In y=3*cos(x)−2 the amplitude is |3|=3 This means the graph will stretch vertically, reaching 3 units above and below the midline.

3. Determine the vertical shift from the constant term. The value −2 indicates a vertical shift downward by 2 units. The new midline (horizontal axis of oscillation) is y=−2

4. Calculate the range using the midline and amplitude. The maximum value is −2+3=1 and the minimum value is −2−3=−5

5. Identify key points over one period [0,2*π] Since there is no horizontal shift or period change, we use the standard x-values:

• At x=0 y=3*cos(0)−2=3*(1)−2=1

• At x=π/2 y=3*cos(π/2)−2=3*(0)−2=−2

• At x=π y=3*cos(π)−2=3*(−1)−2=−5

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

• At x=2*π y=3*cos(2*π)−2=3*(1)−2=1

6. Sketch the curve by plotting these points and connecting them with a smooth wave, ensuring the graph repeats every 2*π units.

Final Answer

y=3*cos(x)−2

The graph is a cosine wave with midline y=−2 amplitude 3 period 2*π and range [−5,1]

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