How do you graph #y=-2+3cos(x-pi)#?

graph{3 cos (x - pi) - 2 [-10, 10, -5, 5]}

To graph the function ( y = -2 + 3\cos(x - \pi) ), follow these steps:

1. Identify the key components of the function:

   • Amplitude: The amplitude of the cosine function is the coefficient of the cosine term, which is 3 in this case.
   • Vertical shift: The vertical shift is the constant term added to the function, which is -2 in this case.
   • Horizontal shift: The phase shift is the value inside the cosine function, which is ( \pi ) in this case.

2. Plot the vertical shift: Since the vertical shift is -2, the graph of the function will be shifted down by 2 units.

3. Determine the period: The period of the cosine function is ( 2\pi ), which means the graph will complete one full cycle every ( 2\pi ) units.

4. Identify the x-intercepts: The x-intercepts occur when the cosine function equals 0. Since the cosine function has a maximum value of 1 and a minimum value of -1, the x-intercepts occur when ( 3\cos(x - \pi) = 0 ), which happens at ( x = \pi, 3\pi, 5\pi, \ldots ).

5. Draw the graph: Use the amplitude, vertical shift, and x-intercepts to sketch the graph of the function. The graph will oscillate between the maximum and minimum values of the cosine function, with a period of ( 2\pi ). The graph will be shifted down by 2 units due to the vertical shift.

6. Label the axes: Label the x-axis with appropriate scales and label the y-axis with the maximum and minimum values of the function.

By following these steps, you can graph the function ( y = -2 + 3\cos(x - \pi) ) accurately.