Precalculus - Lecture 14: Polynomial Functions and Models

Definition of Polynomial

• A polynomial function has the form:
  • ( f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 )
• Coefficients ( a_n, a_{n-1}, \ldots, a_0 ) are real numbers.
• ( n ) is a non-negative integer.

Examples of Polynomials

• Example 1: ( f(x) = 2x^5 - x^4 + 3x^2 - 7 )
  • Coefficients: ( a_5 = 2 ), ( a_4 = -1 ), ( a_3 = 0 ), ( a_2 = 3 ), ( a_1 = 0 ), ( a_0 = -7 )
  • Degree: 5

Characteristics of Polynomials

• Domain: All real numbers.
• Exponents must be non-negative integers (no fractions or negative exponents).

Determining if Functions are Polynomials

1. Example: ( f(x) = x(x+5) )

   • Distribute: ( f(x) = x^2 + 5x )
   • Exponents are integers. Yes, it's a polynomial.
   • Degree: 2

2. Example: ( f(x) = \sqrt{x}(\sqrt{x} - 2) )

   • ( \sqrt{x} = x^{1/2} )
   • Fractional exponent. Not a polynomial.

3. Example: ( f(x) = \frac{x^2 - 5x}{x^3} )

   • Contains negative exponents. Not a polynomial.

4. Example: ( f(x) = -3x^2(x+5)^3 )

   • Exponents are integers. Yes, it's a polynomial.
   • Degree: 5

Power Functions

• Definition: A power function of degree ( n ) is a monomial ( f(x) = ax^n ).

• Even ( n ):

  • Symmetric about the y-axis.
  • Domain: All real numbers.
  • Range: ([0, \infty) ) if ( a > 0 ), ((-\infty, 0] ) if ( a < 0 ).
  • Graph resembles ( y = x^2 ).

• Odd ( n ):

  • Symmetric about the origin.
  • Domain and Range: All real numbers.
  • Graph resembles ( y = x^3 ).

Graphing Power Functions

• Use transformations to graph functions.
• Example: ( f(x) = -x^4 + 3 ):
  • Reflect over x-axis, then shift up 3.
  • Domain: All real numbers, Range: ((-\infty, 3] ).

Zeros and Multiplicity

• Zero ( r ) if ( f(r) = 0 ).
• Multiplicity: Times ( r ) is a zero based on the exponent of the factor.
• Example: ( f(x) = (x-3)^2(x+1) )
  • Zeros: 3 (multiplicity 2), -1 (multiplicity 1).

Forming Polynomials from Zeros

• Use given zeros and degree to construct polynomials.
• Example: Zeros at -2, 2, 3; Degree: 3.
  • Polynomial: ( p(x) = (x+2)(x-2)(x-3) ).

Graph Characteristics

• Even Multiplicity: Graph touches the x-axis.

• Odd Multiplicity: Graph crosses the x-axis.

• Turning Points:

  • Maximum: ( n-1 ) for a degree ( n ) polynomial.

• End Behavior:

  • Resembles the graph of the highest degree term.
  • Example: Polynomial ( f(x) = 2x^5 - x^4 + x^3 - 7 ) resembles ( 2x^5 ).

Full Graph Analysis

1. Find:

   • Real zeros and their multiplicity.
   • X and Y-intercepts.
   • Touch vs. cross at x-intercepts.
   • Maximum turning points.
   • End behavior (using limit notation).

2. Sketch the polynomial function graph.

Practice

• Example: ( f(x) = -x^2(x^2-1)(x+1)^3 ):
  • Factored form: ( f(x) = -x^2(x-1)(x+1)^4 ).
  • Degree: 7.
  • Zeros at: 0 (multiplicity 2), 1 (multiplicity 1), -1 (multiplicity 4).
  • End behavior: Resembles ( -x^7 ).

Conclusion

• Understanding polynomials involves identifying their degree, zeros, and behavior.
• Graphing polynomials require understanding transformations and end behavior.

This wraps up the lecture on polynomial functions and models. Next topic is on rational functions. Keep practicing these concepts for mastery.