Graph y=3cos(x)+2

Problem

y=3*cos(x)+2

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 from the coefficient of the cosine term. In y=3*cos(x)+2 the amplitude is |3|=3 This means the graph stretches vertically, reaching 3 units above and below the midline.

3. Determine the vertical shift from the constant term. The +2 indicates a vertical shift upward by 2 units. The new midline (average value) of the function is y=2

4. Calculate the range by applying the amplitude to the midline. The maximum value is 2 + 3 = 5a*n*d(t)*h*e*m*i*n*i*m*u*m*v*a*l*u*e*i*s() - 3 = -1$.

5. Identify key points over one period [0,2*π] Since there is no horizontal shift or period change, the x-coordinates of the key points remain 0,π/2,π,(3*π)/2,2*π

6. Evaluate the function at these key points:

• At x=0 y=3*cos(0)+2=3*(1)+2=5

• 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=−1

• 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=5

7. Sketch the curve by plotting these points (0,5),(π/2,2),(π,−1),((3*π)/2,2),(2*π,5) and connecting them with a smooth wave.

Final Answer

To graph y=3*cos(x)+2 plot a cosine wave with amplitude 3 midline y=2 period 2*π and range [−1,5]

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