Graph y=4sin(3x-1/3*pi)+1

Problem

y=4*sin(3*x−1/3*π)+1

Solution

1. Identify the general form of the sine function, which is y=A*sin(B*(x−C))+D

2. Factor out the coefficient of x inside the parentheses to find the phase shift.

y=4*sin(3*(x−π/9))+1

3. Determine the amplitude |A| which is the distance from the midline to the peak.

Amplitude=|4|=4

4. Calculate the period using the formula P=(2*π)/B

Period=(2*π)/3

5. Identify the phase shift C which indicates the horizontal displacement.

Phase Shift=π/9* (to the right)

6. Identify the vertical shift D which determines the midline of the graph.

Vertical Shift=1

7. Find the key points by dividing the period into four equal intervals of π/6 and starting from the phase shift π/9

8. Sketch the graph starting at the midline (π/9,1) rising to a maximum at ((5*π)/18,5) returning to the midline at ((4*π)/9,1) dropping to a minimum at ((11*π)/18,−3) and finishing one cycle at ((7*π)/9,1)

Final Answer

y=4*sin(3*x−π/3)+1

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