Polynomial Functions and Their Graphs

Definition and Properties of Polynomial Functions

A polynomial function is a function of the form , where the coefficients are real numbers and is a non-negative integer.

• Degree: The highest power of with a nonzero coefficient.

• Leading Coefficient: The coefficient of the highest degree term.

• End Behavior: Determined by the degree and leading coefficient (see Leading Coefficient Test).

Leading Coefficient Test

The Leading Coefficient Test describes the end behavior of the graph of a polynomial function:

• If is even and , both ends rise.

• If is even and , both ends fall.

• If is odd and , left end falls, right end rises.

• If is odd and , left end rises, right end falls.

Zeros and Multiplicity

• Zero of a polynomial: A value such that .

• Multiplicity: If is a factor, is a zero of multiplicity .

• If is odd, the graph crosses the x-axis at ; if $k$ is even, the graph touches but does not cross.

Turning Points

• A polynomial of degree has at most turning points.

Intermediate Value Theorem

If is a polynomial and and have opposite signs, then there is at least one between and such that .

General Strategy for Graphing Polynomial Functions

• Identify degree and leading coefficient (end behavior).

• Find zeros and their multiplicities.

• Find y-intercept ().

• Plot additional points as needed.

• Sketch the graph, considering turning points and end behavior.

Example

Graph :

• Degree 3 (odd), leading coefficient 1 (positive): left end down, right end up.

• Zeros: , so zeros at (all multiplicity 1).

• Y-intercept: .

Dividing Polynomials

Long Division and the Division Algorithm

Long division is used to divide polynomials, similar to numerical long division.

The Division Algorithm states: For polynomials and , there exist unique polynomials and such that

, where .

Synthetic Division

• Used when dividing by .

• Write coefficients, use as the synthetic divisor, perform the algorithm to find the quotient and remainder.

Remainder Theorem

If is divided by , the remainder is .

Factor Theorem

is a factor of if and only if .

Example

Divide by using synthetic division. The remainder is .

Zeros of Polynomial Functions

Finding Zeros of Degree 3 Polynomials

• Test possible rational zeros using the Rational Root Theorem.

• Once a zero is found, factor it out and solve the reduced polynomial.

Linear Factorization Theorem

If a polynomial of degree has $n$ real or complex zeros (counting multiplicity), it can be written as

Example

Given zeros and , find the polynomial:

. Use to solve for .

Rational Functions and Their Graphs

Definition and Domain

A rational function is a function of the form , where and are polynomials and .

• The domain is all real numbers except where .

Basic Rational Functions

• : Hyperbola, undefined at .

• : Parabola opening upwards, undefined at .

Transformations

• Shifts, stretches, and reflections can be applied to and to graph related functions.

Asymptotes

• Vertical asymptotes: Occur at zeros of (where denominator is zero).

• Horizontal asymptotes: Determined by comparing degrees of and :

Graphing Rational Functions

• Find domain, intercepts, asymptotes.

• Plot key points and sketch the graph, considering asymptotic behavior.

Polynomial and Rational Inequalities

Solving Inequalities

• Express the inequality in the form , , , or .

• Find boundary points (where or is undefined).

• Test intervals between boundary points to determine where the inequality holds.

Example

Solve :

• Boundary points: (numerator zero), (denominator zero).

• Test intervals: , , .

Exponential Functions

Definition and Properties

An exponential function is , where and .

• Domain:

• Range:

• y-intercept:

• Horizontal asymptote:

Natural Exponential Function

The natural exponential function is , where .

Compound Interest Formulas

• For compounding periods per year:

• For continuous compounding:

• Where:

  • = final amount

  • = principal (initial amount)

  • = annual interest rate (decimal)

  • = number of compounding periods per year

  • = number of years

Example

Find the amount after 5 years if , , compounded quarterly ():

For continuous compounding: