To draw the graph of the function \( |f(x)| = \cos x \) for \( x \in [-2\pi, 2\pi] \), we will follow these steps:
### Step 1: Understand the function \( f(x) = \cos x \)
The cosine function oscillates between -1 and 1. It has a period of \( 2\pi \), meaning it repeats every \( 2\pi \) units. The key points for \( \cos x \) in the interval \( [-2\pi, 2\pi] \) are:
- \( \cos(-2\pi) = 1 \)
- \( \cos(-\pi) = -1 \)
- \( \cos(0) = 1 \)
- \( \cos(\pi) = -1 \)
- \( \cos(2\pi) = 1 \)
### Step 2: Plot the graph of \( f(x) = \cos x \)
1. Start by plotting the points mentioned above on the graph.
2. Draw the curve of \( \cos x \) from \( -2\pi \) to \( 2\pi \). The graph will start at \( ( -2\pi, 1) \), go down to \( ( -\pi, -1) \), rise back to \( (0, 1) \), drop down to \( (\pi, -1) \), and finally return to \( (2\pi, 1) \).
### Step 3: Apply the absolute value
Since we are interested in \( |f(x)| = |\cos x| \), we need to consider the absolute values of the points:
- Where \( \cos x \) is positive (i.e., \( [ -\pi/2, \pi/2] \) and \( [3\pi/2, 5\pi/2] \)), the graph remains unchanged.
- Where \( \cos x \) is negative (i.e., \( [ -\pi, 0] \) and \( [\pi, 2\pi] \)), we reflect the graph across the x-axis.
### Step 4: Draw the graph of \( |f(x)| = |\cos x| \)
1. For the intervals where \( \cos x \) is negative, reflect those portions above the x-axis.
2. The resulting graph will have peaks at \( 1 \) and troughs at \( 0 \) instead of \( -1 \).
### Final Graph
The final graph will show:
- Peaks at \( ( -2\pi, 1) \), \( (0, 1) \), and \( (2\pi, 1) \).
- Troughs at \( ( -\pi, 0) \) and \( (\pi, 0) \).
### Summary of the Graph
The graph of \( |f(x)| = \cos x \) will oscillate between 0 and 1, touching the x-axis at \( -\pi \) and \( \pi \), and reaching a maximum of 1 at \( -2\pi \), \( 0 \), and \( 2\pi \).
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