Graph y=cos(x)+3

Problem

y=cos(x)+3

Solution

1. Identify the parent function, which is y=cos(x) The standard cosine curve oscillates between −1 and 1 with a period of 2*π

2. Determine the vertical shift by looking at the constant added to the function. In y=cos(x)+3 the value d=3 indicates a vertical shift upward by 3 units.

3. Find the new midline of the graph. The original midline y=0 moves up to y=3

4. Calculate the range by applying the shift to the original bounds. The minimum value is −1+3=2 and the maximum value is 1 + 3 = 4$.

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

• At x=0 y=cos(0)+3=1+3=4

• At x=π/2 y=cos(π/2)+3=0+3=3

• At x=π y=cos(π)+3=−1+3=2

• At x=(3*π)/2 y=cos((3*π)/2)+3=0+3=3

• At x=2*π y=cos(2*π)+3=1+3=4

6. Sketch the curve by connecting these points with a smooth, wave-like shape, ensuring the graph repeats every 2*π units.

Final Answer

y=cos(x)+3

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