6.1 Part 1 - Graphing Polynomials

Chapter 6: Polynomial Functions

Section 6.1: Introduction to Polynomial Functions

• Focus on polynomial functions, definitions, and methods for solving them.

• Emphasizing the transition from regular functions to polynomial-specific functions.

Definition of Polynomial Functions

• A polynomial is an equation with one variable.

  • Only the variable x is considered.

• General form of a polynomial:

  • $P(x) = a<em>n x^n + a</em>{n-1} x^{n-1} + … + a_0$

  • Where n is a non-negative integer, and each a is a constant (coefficients).

  • The degree of the polynomial is n, and the leading term is a_n x^n.

  • Conditions:

    • No negative exponents allowed.

    • Leading coefficient (a_n) cannot be zero.

Key Characteristics of Polynomials

• Degree: The highest exponent in the polynomial.

• Leading Coefficient: The coefficient of the leading term (the term with the highest exponent).

  • Example: For $3x^2$,

  • Degree = 2,

  • Leading Coefficient = 3.

• Example of higher degree polynomial: $-6x^6$,

  • Degree = 6,

  • Leading Coefficient = -6.

Zeros of a Polynomial

• Zeros (or roots) of a polynomial are values of x for which the polynomial equals zero:

  • If $P(x) = 0$, then x is a zero of the polynomial.

• Various terms used:

  • Zeros

  • Roots

  • Solutions

• Relationship:

  • Finding zeros can be expressed as solving $P(x) = 0$.

• Example of finding zeros:

  • For the polynomial $x^2 + 7x - 8$:

  • Factoring gives: $P(x) = (x + 8)(x - 1)$

  • Thus zeros are: $x = -8$ and $x = 1$.

Verifying Zeros

• Process to verify if a value is a zero:

  • Simply substitute the value into the polynomial and check if it equals zero.

• Example: For $P(x) = -x^3 + 6x^2 - 5x - 12$:

  • Check if x = 4 is a zero:

  • Calculate: $P(4) = -4^3 + 6(4^2) - 5(4) - 12$ = Check if equals zero.

Polynomial Behavior

• Behavior of polynomials can be inferred from their degree and leading coefficient:

  • If leading coefficient is positive, as $x$ approaches ,

  • Left Infinity: P(x) o +

  • Right Infinity: P(x) o +

  • If the exponent is even, the ends of the polynomial will point in the same direction.

  • If the exponent is odd, the ends will point in opposite directions:

  • As x approaches left infinity, y decreases.

  • As x approaches right infinity, y increases.

Graphing Polynomials

• To graph polynomials:

  • Identify intercepts (both x and y).

  • Analyze end behavior based on the degree.

• Example for Cubic Functions:

  • Find x-intercepts using the zero-factor property.

  • An organized approach includes checking several strategic points for accurate sketches.

Example Graphing Problems

• We can analyze polynomial graphs by determining:

  • End behavior based on polynomial degree.

  • Finding x-intercepts through factoring or using the quadratic formula if necessary.

  • Plotting additional points can enhance accuracy in sketches.

Finding Polynomials from Graphs

• Given intercepts of a polynomial, we can construct the polynomial:

  • From found roots $c<em>1, c</em>2, c_3$, the polynomial can be expressed as:

  • $P(x) = a (x - c<em>1)(x - c</em>2)(x - c_3)$

  • Determine the leading coefficient by analyzing the behavior of the graph.

Example: Finding a Polynomial

• For a graph with intercepts at $(-2, 0)$, $(1, 0)$, $(2, 0)$:

  • Polynomial expression:

  • $P(x) = a (x + 2)(x - 1)(x - 2)$

  • Substitute known points to find a.

Summary of Polynomial Characteristics

• Polynomials are defined by their degree and leading coefficients.

• Zeros can be found through factorization or verification.

• Behavior and graphing strategies are central to understanding polynomials effectively in problem-solving contexts.

• Tools such as the zero factor property play critical roles.