Graph y=cos(x-3)

Problem

y=cos(x−3)

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 starts its cycle at a maximum point (0,1)

2. Determine the horizontal shift by analyzing the argument of the cosine function. The expression x−3 indicates a phase shift of 3 units to the right.

3. Locate key points for one period of the shifted graph. Since the original key points occur at x=0,π/2,π,(3*π)/2,2*π the new key points occur at x=3,3+π/2,3+π,3+(3*π)/2,3+2*π

4. Calculate coordinates for the shifted key points.

x=3⇒y=1

x≈4.57⇒y=0

x≈6.14⇒y=−1

x≈7.71⇒y=0

x≈9.28⇒y=1

5. Sketch the curve by plotting these points and drawing a smooth, periodic wave with an amplitude of 1 and a midline at y=0

Final Answer

y=cos(x−3)

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