Responder
To graph \( y = \cos\left(x + \frac{2\pi}{3}\right) \) within one cycle \([0, 2\pi)\), plot the following key points:
- **Maximum**: \( \left(\frac{4\pi}{3}, 1\right) \)
- **Minimum**: \( \left(\frac{\pi}{3}, -1\right) \)
- **X-Intercepts**: \( \left(\frac{5\pi}{6}, 0\right) \) and \( \left(\frac{11\pi}{6}, 0\right) \)
Draw the cosine curve connecting these points, ensuring the graph is shifted left by \( \frac{2\pi}{3} \) and oscillates between \( -1 \) and \( 1 \).
Solución
To graph the trigonometric function \( y = \cos\left(x + \frac{2\pi}{3}\right) \) within one cycle \([0, 2\pi)\), follow these steps to identify and plot the key points: **x-intercepts**, **maxima**, and **minima**.
### 1. **Understanding the Function**
The function \( y = \cos\left(x + \frac{2\pi}{3}\right) \) is a **cosine function** with:
- **Amplitude**: 1 (since there's no coefficient multiplying the cosine)
- **Period**: \( 2\pi \) (standard for cosine)
- **Phase Shift**: \( -\frac{2\pi}{3} \) (shifted **left** by \( \frac{2\pi}{3} \))
- **Vertical Shift**: 0 (no constant added or subtracted)
### 2. **Identifying Key Points**
#### **a. Maxima**
- **Cosine reaches its maximum value of 1** when the argument is \( 0 + 2k\pi \) (where \( k \) is an integer).
- Set \( x + \frac{2\pi}{3} = 0 \):
\[
x = -\frac{2\pi}{3}
\]
- To find within \([0, 2\pi)\), add \( 2\pi \):
\[
x = -\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}
\]
- **Maximum Point**: \( \left(\frac{4\pi}{3}, 1\right) \)
#### **b. Minima**
- **Cosine reaches its minimum value of -1** when the argument is \( \pi + 2k\pi \).
- Set \( x + \frac{2\pi}{3} = \pi \):
\[
x = \pi - \frac{2\pi}{3} = \frac{\pi}{3}
\]
- **Minimum Point**: \( \left(\frac{\pi}{3}, -1\right) \)
#### **c. X-Intercepts**
- **Cosine equals zero** when the argument is \( \frac{\pi}{2} + k\pi \).
1. **First X-Intercept:**
\[
x + \frac{2\pi}{3} = \frac{\pi}{2} \Rightarrow x = \frac{\pi}{2} - \frac{2\pi}{3} = -\frac{\pi}{6}
\]
Adjusting to \([0, 2\pi)\):
\[
x = -\frac{\pi}{6} + 2\pi = \frac{11\pi}{6}
\]
- **First X-Intercept**: \( \left(\frac{11\pi}{6}, 0\right) \)
2. **Second X-Intercept:**
\[
x + \frac{2\pi}{3} = \frac{3\pi}{2} \Rightarrow x = \frac{3\pi}{2} - \frac{2\pi}{3} = \frac{5\pi}{6}
\]
- **Second X-Intercept**: \( \left(\frac{5\pi}{6}, 0\right) \)
### 3. **Plotting the Graph**
Here's a step-by-step guide to sketch the graph:
1. **Draw the Axes:**
- Horizontal axis (\( x \)) from \( 0 \) to \( 2\pi \).
- Vertical axis (\( y \)) from \( -1.5 \) to \( 1.5 \).
2. **Plot the Key Points:**
- **Maximum** at \( \left(\frac{4\pi}{3}, 1\right) \).
- **Minimum** at \( \left(\frac{\pi}{3}, -1\right) \).
- **X-Intercepts** at \( \left(\frac{5\pi}{6}, 0\right) \) and \( \left(\frac{11\pi}{6}, 0\right) \).
3. **Draw the Cosine Curve:**
- Start from the maximum point.
- Decrease to the x-intercept at \( \frac{5\pi}{6} \).
- Reach the minimum at \( \frac{\pi}{3} \).
- Increase to the x-intercept at \( \frac{11\pi}{6} \).
- Return to the maximum at \( \frac{4\pi}{3} \).
### 4. **Visual Representation**
While I can't provide an interactive graph here, below is a **schematic representation** of the graph:
```
y
|
1 | * (4π/3, 1)
| / \
| / \
0 |--------*-----*---------> x
| (5π/6,0) (11π/6,0)
-1| \ /
| \ /
| * (π/3, -1)
|
```
### 5. **Summary of Key Points**
| Type | \( x \)-value | \( y \)-value | Coordinates |
|------------|-------------------------|---------------|-------------------------|
| **Maximum**| \( \frac{4\pi}{3} \) | 1 | \( \left(\frac{4\pi}{3}, 1\right) \) |
| **Minimum**| \( \frac{\pi}{3} \) | -1 | \( \left(\frac{\pi}{3}, -1\right) \) |
| **X-Intercepts** | \( \frac{5\pi}{6} \), \( \frac{11\pi}{6} \) | 0 | \( \left(\frac{5\pi}{6}, 0\right) \), \( \left(\frac{11\pi}{6}, 0\right) \) |
### 6. **Final Notes**
- **Periodicity**: The graph repeats every \( 2\pi \).
- **Phase Shift**: The entire graph is shifted to the **left** by \( \frac{2\pi}{3} \) compared to the standard cosine function \( y = \cos(x) \).
- **Amplitude and Vertical Shift**: Since the amplitude is 1 and there's no vertical shift, the graph oscillates between \( -1 \) and \( 1 \) without moving up or down.
By plotting these points and following the cosine wave's shape, you can accurately sketch the graph of \( y = \cos\left(x + \frac{2\pi}{3}\right) \) within one cycle.
Revisado y aprobado por el equipo de tutoría de UpStudy
Explicar
Simplifique esta solución
