Graphing polynomial functions involves a systematic approach of identifying key features and sketching the curve. Here's a breakdown of the process:
1. Find the Intercepts
- x-intercepts (Roots/Zeros): Set the polynomial function, f(x), equal to zero and solve for x. The solutions are the x-intercepts, where the graph crosses or touches the x-axis.
- y-intercept: Set x = 0 in the polynomial function and solve for y. This gives the point (0, y), where the graph intersects the y-axis.
2. Check for Symmetry (Optional, but helpful)
- Even Function: If f(-x) = f(x), the function is even, and the graph is symmetric about the y-axis.
- Odd Function: If f(-x) = -f(x), the function is odd, and the graph is symmetric about the origin.
- If neither condition is met, the function has no particular symmetry.
3. Determine Behavior at the x-intercepts
- Multiplicity: The multiplicity of a root is the number of times a factor (x - a) appears in the factored form of the polynomial.
- Odd Multiplicity: The graph crosses the x-axis at the x-intercept.
- Even Multiplicity: The graph touches the x-axis and turns around (is tangent to the x-axis) at the x-intercept.
4. Determine End Behavior
The end behavior of a polynomial function is determined by its leading term (the term with the highest degree).
| Leading Term | End Behavior |
|---|---|
| Positive, Even Degree | Both ends point upwards (as x → ±∞, y → ∞) |
| Positive, Odd Degree | Left end points down, right end points up (as x → -∞, y → -∞; as x → ∞, y → ∞) |
| Negative, Even Degree | Both ends point downwards (as x → ±∞, y → -∞) |
| Negative, Odd Degree | Left end points up, right end points down (as x → -∞, y → ∞; as x → ∞, y → -∞) |
5. Find Additional Points (Optional, but recommended)
- Choose x-values between the x-intercepts and calculate the corresponding y-values. This helps to refine the shape of the graph. Consider using the first derivative to find local maxima and minima to further refine your graph.
6. Sketch the Graph
- Plot the intercepts.
- Use the multiplicities of the roots to sketch the behavior at the x-intercepts.
- Use the end behavior to sketch the ends of the graph.
- Connect the points with a smooth curve, ensuring the curve follows the behavior determined in the previous steps.
Example:
Let's graph the polynomial function f(x) = x³ - x
- Intercepts:
- x-intercepts: x³ - x = 0 => x(x² - 1) = 0 => x(x-1)(x+1) = 0. So, x = -1, 0, 1. Points are (-1, 0), (0, 0), (1, 0).
- y-intercept: f(0) = 0³ - 0 = 0. Point is (0, 0).
- Symmetry:
- f(-x) = (-x)³ - (-x) = -x³ + x = -(x³ - x) = -f(x). Therefore, the function is odd and symmetric about the origin.
- Behavior at x-intercepts:
- All roots have a multiplicity of 1 (odd). The graph crosses the x-axis at x = -1, 0, and 1.
- End Behavior:
- Leading term: x³ (positive, odd degree). As x → -∞, y → -∞; as x → ∞, y → ∞. Left end points down, right end points up.
- Sketch: Plot the intercepts, consider the symmetry and end behavior and draw a smooth curve.
By following these steps, you can effectively graph polynomial functions and understand their behavior.
