Graph y=cos(x+pi/4)

Problem

y=cos(x+π/4)

Solution

1. Identify the parent function and its key characteristics. The parent function is y=cos(x) which has a period of 2*π an amplitude of 1 and starts at a maximum point (0,1)

2. Determine the phase shift by looking at the argument of the cosine function. The expression x+π/4 indicates a horizontal shift. Setting x+π/4=0 gives x=−π/4

3. Apply the horizontal shift to the key points of one period. Every xcoordinate of the standard cosine graph is shifted to the left by π/4 units.

4. Calculate the new key points for one cycle. The standard points (0,1) (π/2,0) (π,−1) ((3*π)/2,0) and (2*π,1) become:

(−π/4,1)

(π/4,0)

((3*π)/4,−1)

((5*π)/4,0)

((7*π)/4,1)

5. Sketch the curve through these points. The graph maintains the same wave shape, amplitude, and period as y=cos(x) but starts its cycle at x=−π/4

Final Answer

y=cos(x+π/4)

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