Answer :
Step-by-step explanation:
To graph the equation \( y = x^3 + 1 \) and provide the explanation, follow these steps:
### Explanation
1. **Understand the Function**:
- The given equation \( y = x^3 + 1 \) is a cubic function. The general form of a cubic function is \( y = ax^3 + bx^2 + cx + d \).
- In this specific equation, \( a = 1 \), \( b = 0 \), \( c = 0 \), and \( d = 1 \).
2. **Basic Shape of the Graph**:
- Cubic functions typically have an S-shaped curve.
- Since the coefficient of \( x^3 \) is positive, the graph will increase from left to right, going from negative infinity to positive infinity.
- The "+1" indicates a vertical shift upwards by 1 unit.
3. **Key Points**:
- The y-intercept occurs when \( x = 0 \): \( y = 0^3 + 1 = 1 \). Thus, the graph will pass through (0, 1).
- As \( x \) approaches negative infinity, \( y \) approaches negative infinity.
- As \( x \) approaches positive infinity, \( y \) approaches positive infinity.
4. **Symmetry**:
- The function \( y = x^3 + 1 \) is not symmetric about the y-axis or the origin, but it is symmetric with respect to the point \((0, 1)\) in terms of cubic transformation.
5. **Behavior at Critical Points**:
- Since there are no quadratic or linear terms, there are no inflection points or local maxima/minima, other than the general behavior of cubic functions.
### Graph
Let's plot the graph of \( y = x^3 + 1 \).
```python
import numpy as np
import matplotlib.pyplot as plt
# Define the function
def f(x):
return x**3 + 1
# Generate x values
x = np.linspace(-3, 3, 400)
# Generate y values
y = f(x)
# Create the plot
plt.figure(figsize=(8, 6))
plt.plot(x, y, label='$y = x^3 + 1
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