How do you graph y = cos 2x?

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

Understanding the Function

The equation \( y = \cos(2x) \) is a cosine function with a frequency that is doubled compared to the standard cosine function \( y = \cos(x) \). This means the wave will complete its cycles more quickly.

Key Features to Identify

• Amplitude: The amplitude remains 1, meaning the graph will oscillate between 1 and -1.
• Period: The period of \( y = \cos(2x) \) is calculated as \( \frac{2\pi}{2} = \pi \). This indicates that the graph will complete one full cycle from 0 to \( \pi \).
• Key Points: Important points to plot include:
  • At \( x = 0 \), \( y = 1 \)
  • At \( x = \frac{\pi}{4} \), \( y = 0 \)
  • At \( x = \frac{\pi}{2} \), \( y = -1 \)
  • At \( x = \frac{3\pi}{4} \), \( y = 0 \)
  • At \( x = \pi \), \( y = 1 \)

Steps to Graph

1. Draw the x-axis and y-axis on your graph.
2. Mark the key points calculated above on the graph.
3. Connect these points smoothly to form the cosine wave.
4. Extend the graph in both directions, repeating the wave pattern.

Final Touches

Label your axes and indicate the amplitude and period for clarity. This will help anyone reading your graph understand its characteristics at a glance.