college algebra

tutorial 35: graphs of polynomial functions

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desklearning objectives

after completing this tutorial, you should be able to:

identify a polynomial function.

use the leading coefficient test to find the end behavior of the graph of a given polynomial function.

find the zeros of a polynomial function.

find the multiplicity of a zero and know if the graph crosses the x-axis at the zero or touches the x-axis and turns around at the zero.

know the maximum number of turning points a graph of a polynomial function could have.

graph a polynomial function.

deskintroduction

in this tutorial we will be looking at graphs of polynomial functions. if you need a review on functions, feel free to go to tutorial 30: introduction to functions. if you need a review on polynomials in general, feel free to go to tutorial 6: polynomials. basically, the graph of a polynomial function is a smooth continuous curve. there are several main aspects of this type of graph that you can use to help put the curve together. i will be going over how to use the leading term of your polynomial function to determine the end behavior of its graph. we will also be looking at finding the zeros, aka the x-intercepts, as well as the y-intercept of the graph. if you need a review on intercepts, feel free to go to tutorial 26: equations of lines. another important concept is to know the largest possible number of turning points. this will help you be more accurate in the graph that you draw. that just about covers it. i guess you are ready to get to it.

desk tutorial

polynomial function

a polynomial function is a function that can be written in the form

polynomial, where

coefficientare real numbers and

n is a nonnegative integer.

basically it is a function whose rule is given by a polynomial in one variable.

if you need a review on functions, feel free to go to tutorial 30: introduction to functions. if you need a review on polynomials in general, feel free to go to tutorial 6: polynomials.

an example of a polynomial function is polynomial.

leading term

when the polynomial function is written in standard form,

polynomial,

the leading term is leading term.

in other words, the leading term is the term that the variable has its highest exponent.

the leading term of the function polynomial would be polynomial.

leading coefficient

when the polynomial function is written in standard form,

polynomial,

the leading coefficient is leading coefficient.

basically, the leading coefficient is the coefficient on the leading term.

the leading coefficient of the function polynomial would be - 4.

degree of a term of a polynomial function

the degree of a term of a polynomial function is the exponent on the variable.

degree of a polynomial function

when the polynomial function is written in standard form,

polynomial,

the degree of the polynomial function is n.

the degree of the polynomial is the largest degree of all of its terms.

the degree of the function polynomial would be 7.

the leading coefficient test

there are four cases that go with this test:

given a polynomial function in standard form polynomial:

case 1:

if n is odd and the leading coefficient leading coefficient, is positive, the graph falls to the left and rises to the right:

odd positive

case 2:

if n is odd and the leading coefficient leading coefficient, is negative, the graph rises to the left and falls to the right.

odd negative

case 3:

if n is even and the leading coefficient leading coefficient, is positive, the graph rises to the left and to the right.

even positive

case 4:

if n is even and the leading coefficient leading coefficient, is negative, the graph falls to the left and to the right.

even negative