Introduction to Polynomial Functions

Overview of Polynomial Functions

• Standard Form: A polynomial function is expressed as a monomial or a sum of monomials.
  • Exponents must be whole numbers (positive integers).
  • Coefficients must be real numbers (can be positive, negative, fractions, decimals, or radicals).

Key Components of Polynomial Functions

• Lead Coefficient: The coefficient of the term with the largest exponent.
  • Indicates the direction of the graph depending on whether it is positive or negative.
• Degree: The largest exponent in the polynomial.
  • Classifies the polynomial and influences the shape of its graph.
• Constant Term: A number at the end with no variable.
• Standard Form Arrangement: Terms are organized from largest to smallest exponent, followed by the constant.

Identifying Polynomial Functions

• Ensure all exponents are whole numbers and coefficients are real numbers to be classified as a polynomial function.
• Example Problems:
  1. Reorder exponents to write in standard form. Check for whole number exponents and real number coefficients.
  2. Identify non-polynomial functions due to negative exponents.

Graphing Polynomial Functions

• Direction Changes: The number of directional changes in the graph matches the degree of the polynomial.
  • Degree 1: One direction change, etc.

End Behavior

• Patterns to Recognize:
  • Odd degrees: Graph ends behavior in opposite directions.
  • Even degrees: Graph ends behavior in the same direction.
• Effect of Lead Coefficient:
  • Positive lead coefficient: Graph points up as x approaches positive infinity.
  • Negative lead coefficient: Graph points down as x approaches positive infinity.

Mathematical Notation for End Behavior

• Description:
  • f(x) approaches positive infinity as x approaches positive infinity: Y gets larger as X gets larger.
  • f(x) approaches negative infinity as x approaches negative infinity: Y gets smaller as X gets smaller.

Practicing End Behavior

• Sketch a simple graph to determine direction based on lead coefficient and degree.
• Practice writing end behavior using the described notation.

Evaluation of Polynomial Functions

• Evaluating: Substitute a given value (e.g., x = 3) into the polynomial to find a specific point.
  • Example: f(3) = 162 - 72 + 15 - 7 = 98.

Conclusion

• Practice evaluating polynomial functions and writing end behavior to understand graphs and specific points.