Algebra 3.2 notes

Chapter 3: Polynomial and Rational Functions

Section 3.2: Polynomial Functions and Their Graphs

Objectives

• Identify polynomial functions.

• Recognize characteristics of graphs of polynomial functions.

• Determine end behavior.

• Use factoring to find zeros of polynomial functions.

• Identify zeros and their multiplicities.

• Use the Intermediate Value Theorem.

• Understand the relationship between degree and turning points.

• Graph polynomial functions.

Definition of a Polynomial Function

• A polynomial function of degree n is defined by:

  • Let n be a nonnegative integer

  • Let coefficients be real numbers

  • The leading coefficient is the coefficient of the variable with the highest power.

Graphs of Polynomial Functions

• Graphs of polynomials of degree 2 or higher:

  • Smooth: Contains only rounded curves, no sharp corners.

  • Continuous: Can be drawn without lifting a pencil, no breaks in the graph.

End Behavior of Polynomial Functions

• End behavior refers to the behavior of the graph at the extremes (far left and right).

  • The graph may have increasing/decreasing intervals but will eventually rise/fall without bound.

  • Determined by the sign of the leading coefficient and the degree of the polynomial.

Leading Coefficient Test

• As x increases or decreases without bound, the end behavior of the graph can be explained through:

  • Leading coefficient

  • Degree of polynomial n.

Example: Using the Leading Coefficient Test

• Given a polynomial of degree 4 (even), with a positive leading coefficient:

  • The graph rises on both ends (left and right).

Zeros of Polynomial Functions

• Zeros of the polynomial function (where f(x)=0):

  • Correspond to x-intercepts on the graph.

  • Real roots represent where the graph intersects the x-axis.

Example: Finding Zeros of a Polynomial Function

• To find the zeros, set the polynomial equal to zero and solve:

  • Examine x-intercepts to see how the graph behaves at these points.

Multiplicity and x-Intercepts

• Multiplicity of zeros:

  • Even multiplicity: Graph touches and turns around at the x-axis.

  • Odd multiplicity: Graph crosses the x-axis.

  • Zeros of multiplicity greater than one cause the graph to flatten out near those points.

Example: Finding Zeros and Their Multiplicities

• Identify zeros and determine multiplicities:

  • Example: Some zeros might have multiplicity of 2 or 3.

The Intermediate Value Theorem

• If f is a polynomial function with real coefficients and f(a) and f(b) have opposite signs:

  • There is at least one real root c between a and b.

Example: Using the Intermediate Value Theorem

• Shows that the polynomial has a real zero between given points based on sign changes.

Turning Points of Polynomial Functions

• A polynomial function of degree n can have at most n-1 turning points in its graph.

Graphing a Polynomial Function

Steps include:

1. Use the Leading Coefficient Test for end behavior.

2. Find x-intercepts by solving the polynomial equation.

   • If there is a zero at r:

     • Even multiplicity: graph touches x-axis and turns around.

     • Odd multiplicity: graph crosses x-axis.

     • Multiplicity > 1: graph flattens near the point.

A Strategy for Graphing Polynomial Functions

1. Find the y-intercept by evaluating f(0).

2. Check for symmetry.

   • y-axis or origin symmetry.

3. Maximum turning points can be verified as n-1.

Example: Graphing a Polynomial Function

• Step 1: Determine end behavior based on leading coefficient and degree.

• Step 2: Identify x-intercepts (zeros) and their multiplicities.

• Step 3: Compute the y-intercept.

• Step 4: Assess any symmetry for graphing assistance.

• Step 5: Confirm the graph adheres to maximum turning point rules based on degree.