Learning Objectives
Introduction
Tutorial
A polynomial function is a function that can be written in the form
, where
are real numbers and
n is a nonnegative integer.
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 .
When the polynomial function is written in standard form,
,
the leading term is .
The leading term of the function
would be .
When the polynomial function is written in standard form,
,
the leading coefficient is .
The leading coefficient of the function would be - 4.
When the polynomial function is written in standard form,
,
the degree of the polynomial function is n.
The degree of the function would be 7.
The Leading Coefficient TestIf n is odd AND the leading coefficient , is positive, the graph falls to the left and rises to the right:
If n is odd AND the leading coefficient , is negative, the graph rises to the left and falls to the right.
If n is even AND the leading coefficient ,
is positive, the graph rises to the left and to the
right.
If n is even AND the leading coefficient ,
is negative, the graph falls to the left and to the
right.
If you said ,
you are correct!!
Second question is what is the leading term’s degree?
If you said 3, you are right on!! 3 is the
exponent on the leading
term, which also means it is the degree of the polynomial.
Third question is what is the coefficient on the leading term?
If you said 5, pat yourself on the back!!
Putting this information together with the Leading Coefficient Test we can determine the end behavior of the graph of our given polynomial:
Since the degree of the polynomial, 3, is odd and the leading coefficient, 5, is positive, then the graph of the given polynomial falls to the left and rises to the right.
If you said ,
you are correct!!
Second question is what is the leading term’s degree?
If you said 4, you are right on!! 4 is the
exponent on the leading
term, which also means it is the degree of the polynomial.
Third question is what is the coefficient on the leading term?
If you said -1, pat yourself on the back!!
Putting this information together with the Leading Coefficient Test we can determine the end behavior of the graph of our given polynomial:
Since the degree of the polynomial, 4, is even and the leading coefficient, -1, is negative, then the graph of the given polynomial falls to the left and falls to the right.
If you said ,
you are correct!!
Second question is what is the leading term’s degree?
If you said 5, you are right on!! 5 is the
exponent on the leading
term, which also means it is the degree of the polynomial.
Third question is what is the coefficient on the leading term?
If you said -7, pat yourself on the back!!
Putting this information together with the Leading Coefficient Test we can determine the end behavior of the graph of our given polynomial:
Since the degree of the polynomial, 5, is odd and the leading coefficient, -7, is negative, then the graph of the given polynomial rises to the left and falls to the right.
If you said ,
you are correct!!
Second question is what is the leading term’s degree?
If you said 6, you are right on!! 6 is the
exponent on the leading
term, which also means it is the degree of the polynomial.
Third question is what is the coefficient on the leading term?
If you said 1, pat yourself on the back!!
Putting this information together with the Leading Coefficient Test we can determine the end behavior of the graph of our given polynomial:
Since the degree of the polynomial, 6, is even and the leading coefficient, 1, is positive, then the graph of the given polynomial rises to the left and rises to the right.
In other words it is the x-intercept, where the functional value or y is equal to 0.
If is a factor of a polynomial function f and
is not a factor of f, then r is called a zero of
multiplicity k of f.
The exponent indicates how many times that factor would be written out in the product, this gives us a multiplicity.
If r is a zero
of even multiplicity:
This means the graph touches the x-axis
at r and turns around.
This happens because the sign of f(x) does not change from one side to the other side of r.
If r is a zero
of odd multiplicity:
This means the graph crosses the x-axis
at r.
This happens because the sign of f(x) changes from one side to the other side of r.
If f is a polynomial function of degree n, then
there is at most n - 1 turning points on the graph of f.
Keep in mind that you can have fewer than n - 1 turning points, but it will never exceed n - 1 turning points.
The first factor is 3, which is a constant. Therefore, there are no zeros that go with this factor.
*Solve for x
*x = -1/2 is a
zero
If you said the multiplicity for x =
-1/2 is 4, you are correct!!!! Since the exponent on this
factor
is 4, then its multiplicity is 4.
Does the graph cross the x-axis or touch the x-axis and turn around at the zero x = -1/2?
If you said it touches the x-axis and turns around at the zero x = -1/2, pat yourself on the back!!! It does this because the multiplicity is 4, which is even.
If you said the multiplicity for x =
4 is 3, you are correct!!!! Since the exponent on this factor
is 3, then its multiplicity is 3.
Does the graph cross the x-axis or touch the x-axis and turn around at the zero x = 4?
If you said it crosses the x-axis at the zero x = 4, pat yourself on the back!!! It does this because the multiplicity is 3, which is odd.
If you said the multiplicity for x =
0 is 2, you are correct!!!! Since the exponent on this factor
is 2, then its multiplicity is 2.
Does the graph cross the x-axis or touch the x-axis and turn around at the zero x = 0?
If you said it touches the x-axis and turns around at the zero x = 0, pat yourself on the back!!! It does this because the multiplicity is 2, which is even.
If you said the multiplicity for x =
-3 is 1, you are correct!!!! Since the exponent on this
factor
is 1, then its multiplicity is 1.
Does the graph cross the x-axis or touch the x-axis and turn around at the zero x = -3?
If you said it crosses the x-axis at the zero x = -3, pat yourself on the back!!! It does this because the multiplicity is 1, which is odd.
If you said the multiplicity for x =
3 is 1, you are correct!!!! Since the exponent on this factor
is 1, then its multiplicity is 1.
Does the graph cross the x-axis or touch the x-axis and turn around at the zero x = 3?
If you said it crosses the x-axis at the zero x = 3, pat yourself on the back!!! It does this because the multiplicity is 1, which is odd.
If you need a review on x-intercepts,
feel
free to go to Tutorial 26:
Equations
of Lines.
Keep in mind that when
is a factor of your polynomial and
b) if k is odd, the graph crosses the x-axis at r.
If you need a review on y-intercepts, feel free to go to Tutorial 26: Equations of Lines.
Origin symmetry:
Recall that your function is symmetric about the origin if it is an
odd function. In other words, if
f(-x)
= -f(x),
then your function is symmetric about the origin.
If you need a review on even and odd functions, feel free to go to Tutorial 32: Graphs of Functions, Part II.
The graph of polynomial functions is always a smooth continuous curve.
Do you think that the graph rises or falls to the left and to the right?
Since the degree of the polynomial, 4, is even and the leading coefficient, 1, is positive, then the graph of the given polynomial rises to the left and rises to the right.
Since the multiplicity is 2, which is even, then the graph touches the x-axis and turns around at the zero x = 0.
Since the multiplicity is 1, which is odd, then the graph crosses the x-axis at the zero x = 3.
Since the multiplicity is 1, which is odd, then the graph crosses the x-axis at the zero x = -1.
*Plug in -x for x
*Take the opposite of f(x)
Do you think that the graph rises or falls to the left and to the right?
Since the degree of the polynomial, 3, is odd and the leading coefficient, -2, is negative, then the graph of the given polynomial rises to the left and falls to the right.
Since the multiplicity is 1, which is odd, then the graph crosses the x-axis at the zero x = 0.
Since the multiplicity is 1, which is odd, then the graph crosses the x-axis at the zero x = -1.
Since the multiplicity is 1, which is odd, then the graph crosses the x-axis at the zero x = 1.
*Plug in -x for x
*Take the opposite of f(x)
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1b: Given the polynomial function a) use the Leading Coefficient Test to determine the graph’s end behavior, b) find the x-intercepts (or zeros) and state whether the graph crosses the x-axis or touches the x-axis and turns around at each x-intercept, c) find the y-intercept, d) determine the symmetry of the graph, e) indicate the maximum possible turning points, and f) graph.
Need Extra Help on these Topics?
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on March 14, 2012 by Kim Seward.
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