Polynomial Graphing Calculator

Graph polynomial functions, find roots, critical points, and analyze polynomial behavior

Polynomial Setup

Polynomial Function

P(x) = 1

Coefficients

Evaluate at Point

1.0000

Graph Range

Polynomial Graph

xy-55
Roots
Maximum
Minimum
Inflection
Evaluation Point

Polynomial Analysis

Roots (Zeros)

No real roots found in range

Critical Points

(-4.95, 1)
Inflection Point
(-4.75, 1)
Inflection Point
(-4.55, 1)
Inflection Point
(-4.35, 1)
Inflection Point
(-4.15, 1)
Inflection Point
(-3.95, 1)
Inflection Point
(-3.75, 1)
Inflection Point
(-3.55, 1)
Inflection Point
(-3.35, 1)
Inflection Point
(-3.15, 1)
Inflection Point
(-2.95, 1)
Inflection Point
(-2.75, 1)
Inflection Point
(-2.55, 1)
Inflection Point
(-2.35, 1)
Inflection Point
(-2.15, 1)
Inflection Point
(-1.95, 1)
Inflection Point
(-1.75, 1)
Inflection Point
(-1.55, 1)
Inflection Point
(-1.35, 1)
Inflection Point
(-1.15, 1)
Inflection Point
(-0.95, 1)
Inflection Point
(-0.75, 1)
Inflection Point
(-0.55, 1)
Inflection Point
(-0.35, 1)
Inflection Point
(-0.15, 1)
Inflection Point
(0.05, 1)
Inflection Point
(0.25, 1)
Inflection Point
(0.45, 1)
Inflection Point
(0.65, 1)
Inflection Point
(0.85, 1)
Inflection Point
(1.05, 1)
Inflection Point
(1.25, 1)
Inflection Point
(1.45, 1)
Inflection Point
(1.65, 1)
Inflection Point
(1.85, 1)
Inflection Point
(2.05, 1)
Inflection Point
(2.25, 1)
Inflection Point
(2.45, 1)
Inflection Point
(2.65, 1)
Inflection Point
(2.75, 1)
Inflection Point
(2.85, 1)
Inflection Point
(2.95, 1)
Inflection Point
(3.05, 1)
Inflection Point
(3.15, 1)
Inflection Point
(3.25, 1)
Inflection Point
(3.35, 1)
Inflection Point
(3.45, 1)
Inflection Point
(3.55, 1)
Inflection Point
(3.65, 1)
Inflection Point
(3.75, 1)
Inflection Point
(3.85, 1)
Inflection Point
(3.95, 1)
Inflection Point
(4.05, 1)
Inflection Point
(4.15, 1)
Inflection Point
(4.35, 1)
Inflection Point
(4.55, 1)
Inflection Point
(4.75, 1)
Inflection Point
(4.95, 1)
Inflection Point

End Behavior

As x → -∞: P(x) → -∞
As x → +∞: P(x) → -∞

Derivative

P'(x) = 0

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

P(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

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. 1. Identify degree and leading coefficient
  2. 2. Find roots by solving P(x) = 0
  3. 3. Find critical points by solving P'(x) = 0
  4. 4. Determine end behavior
  5. 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