Sketch the graph of cos 4x.

Answer

To sketch the graph of \( y = \cos(4x) \), we will follow these steps:

### Step 1: Determine the Period
The standard cosine function \( y = \cos(x) \) has a period of \( 2\pi \). For the function \( y = \cos(4x) \), the period is calculated by dividing the standard period by the coefficient of \( x \):

\[ \text{Period} = \frac{2\pi}{4} = \frac{\pi}{2} \]

### Step 2: Identify Key Points
Since the period of \( y = \cos(4x) \) is \( \frac{\pi}{2} \), we will find the key points within one period. The cosine function starts at its maximum value (1) at \( x = 0 \), then goes to zero, reaches its minimum value (-1), and returns to zero, before reaching the maximum again.

The key points in one period \( [0, \frac{\pi}{2}] \) are:
- At \( x = 0 \): \( y = \cos(4 \cdot 0) = \cos(0) = 1 \)
- At \( x = \frac{\pi}{8} \): \( y = \cos(4 \cdot \frac{\pi}{8}) = \cos(\frac{\pi}{2}) = 0 \)
- At \( x = \frac{\pi}{4} \): \( y = \cos(4 \cdot \frac{\pi}{4}) = \cos(\pi) = -1 \)
- At \( x = \frac{3\pi}{8} \): \( y = \cos(4 \cdot \frac{3\pi}{8}) = \cos(\frac{3\pi}{2}) = 0 \)
- At \( x = \frac{\pi}{2} \): \( y = \cos(4 \cdot \frac{\pi}{2}) = \cos(2\pi) = 1 \)

### Step 3: Plot the Points
Now we can plot the points:
- \( (0, 1) \)
- \( \left(\frac{\pi}{8}, 0\right) \)
- \( \left(\frac{\pi}{4}, -1\right) \)
- \( \left(\frac{3\pi}{8}, 0\right) \)
- \( \left(\frac{\pi}{2}, 1\right) \)

### Step 4: Sketch the Graph
Using the plotted points, we can sketch the graph of \( y = \cos(4x) \). The graph will oscillate between 1 and -1, completing one full cycle in the interval \( [0, \frac{\pi}{2}] \). Since the cosine function is periodic, we can extend this pattern to the left and right.

### Step 5: Label the Axes
Make sure to label the x-axis with the key points and the y-axis with the values from -1 to 1.

### Final Graph
The final graph will show a wave-like pattern oscillating between 1 and -1, with a period of \( \frac{\pi}{2} \).

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