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 vertical shift. The constant +1 at the end of the equation indicates a vertical shift upward by 1 unit. The new midline (average value) is y=1

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

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

• At x=0 y=4*cos(0)+1=4*(1)+1=5

• At x=π/2 y=4*cos(π/2)+1=4*(0)+1=1

• At x=π y=4*cos(π)+1=4*(−1)+1=−3

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

• At x=2*π y=4*cos(2*π)+1=4*(1)+1=5

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

Final Answer

To graph y=4*cos(x)+1 plot a cosine wave with midline y=1 amplitude 4 and period 2*π

Range: *[−3,5]

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