How do you graph #y = 1/2 cos 2x #?

To graph ( y = \frac{1}{2} \cos(2x) ):

1. Identify key points and intervals:
   • The amplitude of the cosine function is ( \frac{1}{2} ), indicating that the graph oscillates between ( \frac{1}{2} ) and ( -\frac{1}{2} ).
   • The period of the function is ( \frac{2\pi}{2} = \pi ), meaning the graph repeats every ( \pi ) units.
   • The phase shift is ( \frac{\pi}{2} ) to the right.
2. Plot key points:
   • Start with the maximum point at ( \left(\frac{\pi}{2}, \frac{1}{2}\right) ).
   • Next, plot the x-intercepts at ( x = \frac{\pi}{4} ) and ( x = \frac{3\pi}{4} ), corresponding to the points where the cosine function crosses the x-axis.
   • Finally, plot the minimum point at ( \left(\frac{3\pi}{2}, -\frac{1}{2}\right) ).
3. Draw the graph:
   • Use the characteristics of the cosine function to sketch the graph smoothly between the key points.
   • The graph starts at a maximum, then decreases to a minimum, and then increases again.
   • Repeat the pattern for each period, maintaining the amplitude, period, and phase shift.

By following these steps, you can accurately graph the function ( y = \frac{1}{2} \cos(2x) ) on a coordinate plane.