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

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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 $x \to \pm \infty, f (x) \to - \infty$ because of the negative coefficient.

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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.

\begin{matrix} A (w) & = & A (r (w)) \\ = & A (24 + 8 w) \\ = & π {(24 + 8 w)}^{2} \end{matrix}

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.

A General Note

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_{1} x + 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 other than zero. Each expression $a_{i} x^{i}$ is a term of a polynomial function .

Identifying polynomial functions

Which of the following are polynomial functions?

\begin{matrix} f (x) & = & 2 x^{3} \cdot 3 x + 4 \\ g (x) & = & - x (x^{2} - 4) \\ h (x) & = & 5 \sqrt{x + 2} \end{matrix}

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.

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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.

A General Note

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.

How To

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

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

Questions & Answers

if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand

Syamthanda Reply

hey , can you please explain oxidation reaction & redox ?

Boitumelo Reply

hey , can you please explain oxidation reaction and redox ?

Boitumelo

for grade 12 or grade 11?

Sibulele

the value of V1 and V2

Tumelo Reply

advantages of electrons in a circuit

Rethabile Reply

we're do you find electromagnetism past papers

Ntombifuthi

what a normal force

Tholulwazi Reply

it is the force or component of the force that the surface exert on an object incontact with it and which acts perpendicular to the surface

Sihle

what is physics?

Petrus Reply

what is the half reaction of Potassium and chlorine

Anna Reply

how to calculate coefficient of static friction

Lisa Reply

how to calculate static friction

Lisa

How to calculate a current

Tumelo

how to calculate the magnitude of horizontal component of the applied force

Mogano

How to calculate force

Monambi

a structure of a thermocouple used to measure inner temperature

Anna Reply

a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4

Amahle Reply

How is energy being used in bonding?

Raymond Reply

what is acceleration

Syamthanda Reply

a rate of change in velocity of an object whith respect to time

Khuthadzo

how can we find the moment of torque of a circular object

Kidist

Acceleration is a rate of change in velocity.

Justice

t =r×f

Khuthadzo

how to calculate tension by substitution

Precious Reply

Shongi

Leago

use fnet method. how many obects are being calculated ?

Khuthadzo

khuthadzo hii

Hulisani

how to calculate acceleration and tension force

Lungile Reply

you use Fnet equals ma , newtoms second law formula

Masego

please help me with vectors in two dimensions

Mulaudzi Reply

how to calculate normal force

Mulaudzi

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Source: OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6

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