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

Question

Emery Adkins · Accepted Answer

As below.

Standard form of cosine function is #y = A cos (Bx - C) + D# #"Given : " y = 3 cos (x - pi) - 2# #A = 3, B = 1, C = pi, D = -2# #"Amplitude " = |A| = 3# #"Period " = (2pi) / |B| = (2pi) / 1 = 2pi# #"Phase Shift " = -C / B = -pi / 1 = -pi, pi " to the LEFT"# #"Vertical Shift " = D = -2# graph{3 cos (x - pi) - 2 [-10, 10, -5, 5]}

Quinn Anderson · Answer

undefined 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.