Graph y=sin(x+3)

Problem

y=sin(x+3)

Solution

1. Identify the parent function and its properties. The base function is y=sin(x) which has a period of 2*π an amplitude of 1 and passes through the origin (0,0)

2. Determine the phase shift by looking at the argument of the sine function. The expression x+3 indicates a horizontal shift. Setting x+3=0 gives x=−3

3. Apply the horizontal shift to the key points of the parent sine wave. Every point (x,y) on the graph of y=sin(x) is shifted 3 units to the left to become (x−3,y)

4. Calculate key points for one period. The starting point shifts from (0,0) to (−3,0) The peak shifts from (π/2,1) to (π/2−3,1) The midline crossing shifts from (π,0) to (π−3,0) The trough shifts from ((3*π)/2,−1) to ((3*π)/2−3,−1) The period ends at (2*π−3,0)

5. Sketch the curve through these points. The graph maintains the same amplitude of 1 and period of 2*π oscillating between y=1 and y=−1

Final Answer

y=sin(x+3)

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