Section 1.9: Proportionality, Power Functions, And...

Section 1.9: Proportionality, Power Functions, AndPolynomials

We have already studied:

I Linear functionsI Exponential functionsI Natural logarithm

In this section, we will study a few more class of functions:I ProportionalityI Power functionsI Polynomial functions

Section 1.9: Proportionality, Power Functions, AndPolynomials

We have already studied:I Linear functions

I Exponential functionsI Natural logarithm

In this section, we will study a few more class of functions:I ProportionalityI Power functionsI Polynomial functions

Section 1.9: Proportionality, Power Functions, AndPolynomials

We have already studied:I Linear functionsI Exponential functions

I Natural logarithmIn this section, we will study a few more class of functions:I ProportionalityI Power functionsI Polynomial functions

Section 1.9: Proportionality, Power Functions, AndPolynomials

We have already studied:I Linear functionsI Exponential functionsI Natural logarithm

In this section, we will study a few more class of functions:I ProportionalityI Power functionsI Polynomial functions

Section 1.9: Proportionality, Power Functions, AndPolynomials

We have already studied:I Linear functionsI Exponential functionsI Natural logarithm

In this section, we will study a few more class of functions:

I ProportionalityI Power functionsI Polynomial functions

Section 1.9: Proportionality, Power Functions, AndPolynomials

We have already studied:I Linear functionsI Exponential functionsI Natural logarithm

In this section, we will study a few more class of functions:I Proportionality

I Power functionsI Polynomial functions

Section 1.9: Proportionality, Power Functions, AndPolynomials

We have already studied:I Linear functionsI Exponential functionsI Natural logarithm

In this section, we will study a few more class of functions:I ProportionalityI Power functions

I Polynomial functions

Section 1.9: Proportionality, Power Functions, AndPolynomials

We have already studied:I Linear functionsI Exponential functionsI Natural logarithm

In this section, we will study a few more class of functions:I ProportionalityI Power functionsI Polynomial functions

Proportionality

DefinitionWe say y is (directly) proportional to x if there is a nonzeroconstant k such that

y = kx

This k is called the constant of proportionality.

. . . proportional to . . . ≈ . . . a (nonzero) constant multiple of . . .

Example“y is proportional to x” means exactly y = kx for some k , 0.

Proportionality

DefinitionWe say y is (directly) proportional to x if there is a nonzeroconstant k such that

y = kx

This k is called the constant of proportionality.

. . . proportional to . . . ≈ . . . a (nonzero) constant multiple of . . .

Example“y is proportional to x” means exactly y = kx for some k , 0.

Proportionality

DefinitionWe say y is (directly) proportional to x if there is a nonzeroconstant k such that

y = kx

This k is called the constant of proportionality.

. . . proportional to . . . ≈ . . . a (nonzero) constant multiple of . . .

Example“y is proportional to x” means exactly y = kx for some k , 0.

ExampleThe heart mass of a mammal is proportional to its body mass.Write a formula for heart mass, H, as a function of body mass, B.

Solution:H = kB

If a human with a body mass of 70kg has a heart mass of0.42kg, find the constant of proportionality.Solution:

0.42 = k · (70)

k =0.4270

= 0.006

ExampleThe heart mass of a mammal is a constant multiple of its bodymass. Write a formula for heart mass, H, as a function of bodymass, B.

Solution:H = kB

If a human with a body mass of 70kg has a heart mass of0.42kg, find the constant of proportionality.Solution:

0.42 = k · (70)

k =0.4270

= 0.006

ExampleThe heart mass of a mammal is a constant multiple of its bodymass. Write a formula for heart mass, H, as a function of bodymass, B.Solution:

H = kB

If a human with a body mass of 70kg has a heart mass of0.42kg, find the constant of proportionality.Solution:

0.42 = k · (70)

k =0.4270

= 0.006

ExampleThe heart mass of a mammal is a constant multiple of its bodymass. Write a formula for heart mass, H, as a function of bodymass, B.Solution:

H = kB

If a human with a body mass of 70kg has a heart mass of0.42kg, find the constant of proportionality.

Solution:

0.42 = k · (70)

k =0.4270

= 0.006

ExampleThe heart mass of a mammal is a constant multiple of its bodymass. Write a formula for heart mass, H, as a function of bodymass, B.Solution:

H = kB

If a human with a body mass of 70kg has a heart mass of0.42kg, find the constant of proportionality.Solution:

0.42 = k · (70)

k =0.4270

= 0.006

ExampleThe area A of a circular disk is proportional to the square of itsradius r. Write a formula for A as a function of r.

Solution:A = kr2

Indeed, we know that this constant of proportionality is thefamous constant π.

ExampleThe area A of a circular disk is a constant multiple of the squareof its radius r. Write a formula for A as a function of r.

Solution:A = kr2

Indeed, we know that this constant of proportionality is thefamous constant π.

ExampleThe area A of a circular disk is a constant multiple of the squareof its radius r. Write a formula for A as a function of r.Solution:

A = kr2

Indeed, we know that this constant of proportionality is thefamous constant π.

ExampleThe area A of a circular disk is a constant multiple of the squareof its radius r. Write a formula for A as a function of r.Solution:

A = kr2

Indeed, we know that this constant of proportionality is thefamous constant π.

Inverse Proportionality

DefinitionSimilarly, we say y is inversely proportional to x if there is anonzero constant k such that

y =kx

I.e.,x · y = k

So we can also say two quantities are inversely proportional iftheir product is a (nonzero) constant.

Inverse Proportionality

DefinitionSimilarly, we say y is inversely proportional to x if there is anonzero constant k such that

y =kx

I.e.,x · y = k

So we can also say two quantities are inversely proportional iftheir product is a (nonzero) constant.

ExampleThe time, T, taken for a journey is inversely proportional to thespeed of travel v. Write a formula for T as a function of v.

Solution:T =

kv

for some k , 0.

ExampleThe amount G of gasoline one can purchase with $20 isinversely proportional to the price P of each gallon of gasoline.Write a formula for G as a function of P.Solution:

G =kP

for some k , 0.

ExampleThe time, T, taken for a journey is inversely proportional to thespeed of travel v. Write a formula for T as a function of v.Solution:

T =kv

for some k , 0.

ExampleThe amount G of gasoline one can purchase with $20 isinversely proportional to the price P of each gallon of gasoline.Write a formula for G as a function of P.Solution:

G =kP

for some k , 0.

ExampleThe time, T, taken for a journey is inversely proportional to thespeed of travel v. Write a formula for T as a function of v.Solution:

T =kv

for some k , 0.

ExampleThe amount G of gasoline one can purchase with $20 isinversely proportional to the price P of each gallon of gasoline.Write a formula for G as a function of P.

Solution:G =

kP

for some k , 0.

ExampleThe time, T, taken for a journey is inversely proportional to thespeed of travel v. Write a formula for T as a function of v.Solution:

T =kv

for some k , 0.

ExampleThe amount G of gasoline one can purchase with $20 isinversely proportional to the price P of each gallon of gasoline.Write a formula for G as a function of P.Solution:

G =kP

for some k , 0.

Power Function

DefinitionWe say that f (x) is a power function of x if f (x) is proportionalto a constant power of x. I.e.

f (x) = k · xp

for some constants k, p , 0.

Example

5x3 2x100 3x12 4x1.349 9xπ

are all power functions of x

Power Function

DefinitionWe say that f (x) is a power function of x if f (x) is proportionalto a constant power of x. I.e.

f (x) = k · xp

for some constants k, p , 0.

Example

5x3 2x100 3x12 4x1.349 9xπ

are all power functions of x

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4

power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x

= 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x

= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3

= 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x

not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1

not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x

= 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3

= 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2

=59

x−2 power function: k =59

, p = −2

Which of the following functions are power functions of x?

f (x) = 2x4 power function: k = 2, p = 4

f (x) = x = 1x1 power function: k = 1, p = 1

f (x) =1x= 1x−1 power function: k = 1, p = −1

f (x) =5x3 = 5x−3 power function: k = 5, p = −3

f (x) = 2x not a power function

f (x) = 2x4 + 1 not a power function

f (x) = 3√

x = 3x1/2 power function: k = 3, p = 1/2

f (x) = (3√

x)3 = 27x3/2 power function: k = 27, p = 3/2

f (x) =5

9x2 =59

x−2 power function: k =59

, p = −2

Graph of Power Function

Blackboard

Polynomials

DefinitionSums of power functions with non-negative integer exponentsare called polynomials.

Functions given by polynomials arecalled polynomial functions.Polynomial functions are of the form

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

in which n is a non-negative integer, called degree and an is anon-zero number called leading coefficients.We call the term anxn the leading term.

Polynomials

DefinitionSums of power functions with non-negative integer exponentsare called polynomials. Functions given by polynomials arecalled polynomial functions.

Polynomial functions are of the form

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

in which n is a non-negative integer, called degree and an is anon-zero number called leading coefficients.We call the term anxn the leading term.

Polynomials

DefinitionSums of power functions with non-negative integer exponentsare called polynomials. Functions given by polynomials arecalled polynomial functions.Polynomial functions are of the form

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

in which n is a non-negative integer, called degree and an is anon-zero number called leading coefficients.We call the term anxn the leading term.

Polynomials

DefinitionSums of power functions with non-negative integer exponentsare called polynomials. Functions given by polynomials arecalled polynomial functions.Polynomial functions are of the form

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

in which n is a non-negative integer, called degree and an is anon-zero number called leading coefficients.

We call the term anxn the leading term.

Polynomials

DefinitionSums of power functions with non-negative integer exponentsare called polynomials. Functions given by polynomials arecalled polynomial functions.Polynomial functions are of the form

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

in which n is a non-negative integer, called degree and an is anon-zero number called leading coefficients.We call the term anxn the leading term.

Polynomials of low degrees have special names:degree 1 linear

degree 2 quadraticdegree 3 cubicdegree 4 quarticdegree 5 quintic

The study of polynomial function and equations is almost asold as mathematics itself. The quadratic formula, for example,was found in some Babylonian clay tablets (around 2000BC).

Polynomials of low degrees have special names:degree 1 lineardegree 2 quadratic

degree 3 cubicdegree 4 quarticdegree 5 quintic

The study of polynomial function and equations is almost asold as mathematics itself. The quadratic formula, for example,was found in some Babylonian clay tablets (around 2000BC).

Polynomials of low degrees have special names:degree 1 lineardegree 2 quadraticdegree 3 cubic

degree 4 quarticdegree 5 quintic

The study of polynomial function and equations is almost asold as mathematics itself. The quadratic formula, for example,was found in some Babylonian clay tablets (around 2000BC).

Polynomials of low degrees have special names:degree 1 lineardegree 2 quadraticdegree 3 cubicdegree 4 quartic

degree 5 quinticThe study of polynomial function and equations is almost asold as mathematics itself. The quadratic formula, for example,was found in some Babylonian clay tablets (around 2000BC).

Polynomials of low degrees have special names:degree 1 lineardegree 2 quadraticdegree 3 cubicdegree 4 quarticdegree 5 quintic

The study of polynomial function and equations is almost asold as mathematics itself. The quadratic formula, for example,was found in some Babylonian clay tablets (around 2000BC).

Polynomials of low degrees have special names:degree 1 lineardegree 2 quadraticdegree 3 cubicdegree 4 quarticdegree 5 quintic

The study of polynomial function and equations is almost asold as mathematics itself. The quadratic formula, for example,was found in some Babylonian clay tablets (around 2000BC).

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1

polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2

f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x

polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1

not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1

polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4

polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3

f (x) = 2√

x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1

not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7

polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5

f (x) = 7 polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7

polynomial function: deg = 0, LC = 7

f (x) = anxn + an−1xn−1 + · · ·+ a1x1 + a0

ExampleWhich of the following functions are polynomial functions?

f (x) = 2x2 − 3x + 1 polynomial function: deg = 2, LC = 2f (x) = x polynomial function: deg = 1, LC = 1

f (x) = 2x−2 − 3x + 1 not a polynomial function

f (x) = 2x2 − 3x4 + 1 polynomial function: deg = 4, LC = −3

f (x) = x4 − 3x2 − x4 polynomial function: deg = 2, LC = −3f (x) = 2

√x − 3x + 1 not a polynomial function

f (x) = 5x7 polynomial function: deg = 7, LC = 5f (x) = 7 polynomial function: deg = 0, LC = 7

End Behavior of Polynomial Functions

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