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    5.1 part 1 5.1 part 1 Presentation Transcript

    • Chapter 5: Polynomials and Polynomial Functions 5.1: Polynomial Functions
    • Definitions
      • A monomial is a real number, a variable, or a product of a real number and one or more variables with whole number exponents.
        • Examples:
      • The degree of a monomial in one variable is the exponent of the variable.
    • Definitions
      • A polynomial is a monomial or a sum of monomials.
        • Example:
      • The degree of a polynomial in one variable is the greatest degree among its monomial terms.
        • Example:
    • Definitions
      • A polynomial function is a polynomial of the variable x .
        • A polynomial function has distinguishing “behaviors”
          • The algebraic form tells us about the graph
          • The graph tells us about the algebraic form
    • Definitions
      • The standard form of a polynomial function arranges the terms by degree in descending order
        • Example:
    • Definitions
      • Polynomials are classified by degree and number of terms.
        • Polynomials of degrees zero through five have specific names and polynomials with one through three terms also have specific names.
      Degree Name 0 Constant 1 Linear 2 Quadratic 3 Cubic 4 Quartic 5 Quintic Number of Terms Name 1 Monomial 2 Binomial 3 Trinomial 4+ Polynomial with ___ terms
    • Example
      • Write each polynomial in standard form. Then classify it by degree and by number of terms.
    • Example
      • Write each polynomial in standard form. Then classify it by degree and by number of terms.
    • Polynomial Behavior
      • The degree of a polynomial function
        • Affects the shape of its graph
        • Determines the number of turning points (places where the graph changes direction)
        • Affects the end behavior (the directions of the graph to the far left and to the far right)
    • Polynomial Behavior
      • The graph of a polynomial function of degree n has at most n – 1 turning points.
        • Odd Degree = even number of turning points
        • Even Degree = odd number of turning points
      • Think about this:
        • If a polynomial has degree 2, how many turning points can it have?
        • If a polynomial has degree 3, how many turning points can it have?
    • Polynomial Behavior
      • End behavior is determined by the leading term
    • Polynomial Behavior Examples
    • Example
      • Determine the end behavior of the graph of each polynomial function.
    • Example
      • Determine the end behavior of the graph of each polynomial function.
    • Increasing and Decreasing
      • Remember: We read from left to right!
      • A function is increasing when the y-values increase as the x-values increase
      • A function is decreasing when the y-values decrease as the x-values increase
    • Example: Identify the parts of the graph that are increasing or decreasing
    • Example: Identify the parts of the graph that are increasing or decreasing
    • Homework
      • P285 #8 – 31all