Polynomial Graphing Calculator
Graph polynomial functions, find roots, critical points, and analyze polynomial behavior
Polynomial Setup
Polynomial Function
Coefficients
Evaluate at Point
Graph Range
Polynomial Graph
Polynomial Analysis
Roots (Zeros)
No real roots found in range
Critical Points
End Behavior
Derivative
Example Polynomials
Polynomial Properties
Degree
Highest power of x: 2
Leading Coefficient
0
Constant Term
1
Calculator Tips
Higher degree = more complex curves
Leading coefficient determines end behavior
Roots are where the graph crosses x-axis
Critical points show peaks and valleys
Understanding Polynomial Functions
What are Polynomials?
Polynomials are algebraic expressions consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.
General Form
Key Terms
- Degree: Highest power of the variable
- Leading Coefficient: Coefficient of highest degree term
- Root/Zero: Value where P(x) = 0
- Critical Point: Where P'(x) = 0
End Behavior Rules
Even Degree
Positive leading coefficient: both ends go to +∞
Negative leading coefficient: both ends go to -∞
Odd Degree
Positive leading coefficient: left to -∞, right to +∞
Negative leading coefficient: left to +∞, right to -∞
Analysis Steps
- 1. Identify degree and leading coefficient
- 2. Find roots by solving P(x) = 0
- 3. Find critical points by solving P'(x) = 0
- 4. Determine end behavior
- 5. Plot key points and sketch curve
Common Polynomial Types
Linear (Degree 1)
P(x) = ax + b
Straight line, one root
Quadratic (Degree 2)
P(x) = ax² + bx + c
Parabola, up to 2 roots
Cubic (Degree 3)
P(x) = ax³ + bx² + cx + d
S-shaped curve, up to 3 roots
Real-World Applications
Physics
Trajectory calculations, motion under gravity
Economics
Cost functions, profit optimization
Engineering
Signal processing, curve fitting