To graph the trigonometric function \( y = \frac{\cos(2x)}{3} - 4 \), we'll first consider the properties of the cosine function.
The cosine function oscillates between -1 and 1 over a period of \( 2\pi \). However, since we have a coefficient of 2 in front of \( x \), the period is halved to \( \pi \). Additionally, the amplitude is modified by the coefficient 3, stretching it vertically. Finally, we subtract 4, which shifts the graph downward by 4 units.
Given these considerations, let's graph one cycle of the function:
1. **Horizontal Scale**: Since the period is \( \pi \), we can mark the x-axis in increments of \( \frac{\pi}{2} \).
2. **Vertical Scale**: The amplitude of the function is \( \frac{1}{3} \), so we can mark the y-axis in increments of \( \frac{1}{3} \).
3. **Origin**: The function's baseline is shifted downward by 4 units, so the origin will be at (-4, 0).
4. **Shape**: Since cosine starts at its maximum value of 1, we'll begin at the point (-4, 1). Then, we'll follow the cosine curve for one period, which ends at (-2, -1).
Here's the graph of one cycle of the function:
```
|
5 -
|
4 -
|
3 -
|
2 -
|
1 - *
| * \
0 - * / \
| * / \
-1 ---------------------
0 π/2 π 3π/2 2π
```
This graph represents one cycle of the function \( y = \frac{\cos(2x)}{3} - 4 \).
