Graph y=3cos(x)+1

Problem

y=3*cos(x)+1

Solution

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

2. Determine the amplitude. The coefficient of the cosine term is 3 so the amplitude is |3|=3 This means the graph stretches vertically by a factor of 3

3. Identify the vertical shift. The constant term +1 indicates a vertical shift upward by 1 unit. The midline of the function moves from y=0 to y=1

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

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

• At x=0 y=3*cos(0)+1=3*(1)+1=4

• At x=π/2 y=3*cos(π/2)+1=3*(0)+1=1

• At x=π y=3*cos(π)+1=3*(−1)+1=−2

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

• At x=2*π y=3*cos(2*π)+1=3*(1)+1=4

6. Sketch the curve by plotting these key points and connecting them with a smooth wave, repeating the pattern every 2*π units.

Final Answer

y=3*cos(x)+1

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