Understanding the Graph Behavior of f(x) = 2x3 + 26x + 24
The behavior of the graph of a polynomial function, such as f(x) = 2x3 + 26x + 24, can be analyzed by examining various characteristics including:
1. End Behavior
- As x approaches +∞ (positive infinity), f(x) approaches +∞.
- As x approaches -∞ (negative infinity), f(x) approaches -∞.
2. Critical Points
The critical points, where the function’s derivative is zero or undefined, will help identify local maxima and minima. To find critical points:
- Calculate the derivative of the function: f'(x) = 6x2 + 26.
- Setting f'(x) = 0 leads to critical points, which will give insights on increasing or decreasing behavior.
3. Inflection Points
Inflection points occur where the second derivative changes sign:
- Calculate the second derivative: f”(x) = 12x.
- Setting f”(x) = 0 gives x = 0, indicating a change in concavity.
4. Intercepts
- x-intercept: Set f(x) = 0 to find x-intercepts.
- y-intercept: Evaluate f(0) = 24 to find y-intercept at (0, 24).
Graph Summary Table
| Feature | Description |
|---|---|
| End Behavior | Rises to +∞ as x → +∞, falls to -∞ as x → -∞. |
| Critical Points | None, as the derivative does not equal zero; only the stationary point at x = 0. |
| Inflection Points | x = 0 changes concavity from up to down. |
| X-intercepts | To be determined by solving f(x) = 0. |
| Y-intercept | (0, 24). |
By analyzing these features, you gain insights into the behavior and characteristics of the polynomial function, allowing for a comprehensive understanding of its graph.