1. Home
  2. Precalculus
  3. Polynomial and rational functions
  4. Power functions and polynomial

3.3 Power functions and polynomial functions  (Page 3/19)

<< Chapter < Page
  Precalculus     Page 3 / 19
Chapter >> Page >

Describe in words and symbols the end behavior of   f ( x ) = − 5 x 4 .

As   x   approaches positive or negative infinity,   f ( x )   decreases without bound: as   x → ± ∞ ,   f ( x ) → − ∞   because of the negative coefficient.

Got questions? Get instant answers now!

Identifying polynomial functions

An oil pipeline bursts in the Gulf of Mexico, causing an oil slick in a roughly circular shape. The slick is currently 24 miles in radius, but that radius is increasing by 8 miles each week. We want to write a formula for the area covered by the oil slick by combining two functions. The radius   r   of the spill depends on the number of weeks   w   that have passed. This relationship is linear.

r ( w ) = 24 + 8 w

We can combine this with the formula for the area   A   of a circle.

A ( r ) = π r 2

Composing these functions gives a formula for the area in terms of weeks.

A ( w ) = A ( r ( w ) )                                           = A ( 24 + 8 w )                                               = π ( 24 + 8 w ) 2

Multiplying gives the formula.

A ( w ) = 576 π + 384 π w + 64 π w 2

This formula is an example of a polynomial function . A polynomial function consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.

Polynomial functions

Let   n   be a non-negative integer. A polynomial function    is a function that can be written in the form

f ( x ) = a n x n + ... + a 2 x 2 + a 1 x + a 0

This is called the general form of a polynomial function. Each   a i   is a coefficient and can be any real number. Each product   a i x i   is a term of a polynomial function    .

Identifying polynomial functions

Which of the following are polynomial functions?

f ( x ) = 2 x 3 ⋅ 3 x + 4 g ( x ) = − x ( x 2 − 4 ) h ( x ) = 5 x + 2

The first two functions are examples of polynomial functions because they can be written in the form   f ( x ) = a n x n + ... + a 2 x 2 + a 1 x + a 0 ,   where the powers are non-negative integers and the coefficients are real numbers.

  • f ( x )   can be written as   f ( x ) = 6 x 4 + 4.
  • g ( x )   can be written as   g ( x ) = − x 3 + 4 x .
  • h ( x )   cannot be written in this form and is therefore not a polynomial function.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Identifying the degree and leading coefficient of a polynomial function

Because of the form of a polynomial function, we can see an infinite variety in the number of terms and the power of the variable. Although the order of the terms in the polynomial function is not important for performing operations, we typically arrange the terms in descending order of power, or in general form. The degree    of the polynomial is the highest power of the variable that occurs in the polynomial; it is the power of the first variable if the function is in general form. The leading term    is the term containing the highest power of the variable, or the term with the highest degree. The leading coefficient    is the coefficient of the leading term.

Terminology of polynomial functions

We often rearrange polynomials so that the powers are descending.

Diagram to show what the components of the leading term in a function are. The leading coefficient is a_n and the degree of the variable is the exponent in x^n. Both the leading coefficient and highest degree variable make up the leading term. So the function looks like f(x)=a_nx^n +…+a_2x^2+a_1x+a_0.

When a polynomial is written in this way, we say that it is in general form.

Given a polynomial function, identify the degree and leading coefficient.

  1. Find the highest power of   x   to determine the degree function.
  2. Identify the term containing the highest power of   x   to find the leading term.
  3. Identify the coefficient of the leading term.

Questions & Answers

how to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
what is isotopes
Nangun Reply
nuclei having the same Z and different N s
AI-Robot
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 11

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask