Graph y=cos(x+3)

Problem

y=cos(x+3)

Solution

1. Identify the parent function and its properties. The parent function is y=cos(x) which has an amplitude of 1 a period of 2*π and starts 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.

3. Calculate the shift value by setting the argument to zero.

x+3=0

x=−3

This means the graph is shifted 3 units to the left.

4. Identify key points for one period of the shifted graph. Since the period is 2*π≈6.28 the cycle starts at x=−3 and ends at x=−3+2*π≈3.28

• Maximum: (−3,1)

• Intercept: (−3+π/2,0)≈(−1.43,0)

• Minimum: (−3+π,−1)≈(0.14,−1)

• Intercept: (−3+(3*π)/2,0)≈(1.71,0)

• Maximum: (−3+2*π,1)≈(3.28,1)

5. Sketch the curve by plotting these key 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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