WARM UP EXERCSE. 2-1 Polynomials and Rational Functions

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WARM UP EXERCSE

Roots, zeros, and x-intercepts.

f (x) = x

²

! 25

f (x) = x

³

! 25x f (x) = x

²

+ 25

f (x) = polynomial, f (a) = 0 ! f (x) = (x - a)g(x)

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§ 2-1 Polynomials and Rational Functions

Students will learn about:

•Polynomial functions

–Behavior & graphs –Root approximation

•Rational functions:

–Behavior & graphs

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Examples

( )

3

27 f x = ! x x

5 3

( ) 5 4 1

f x = x ! x + x +

f (x) = (x ! 3)³+ 2

=

4 2

( ) 6

f x = x ! x

f (x) = a

_n

x

ⁿ

+ a

_n_!1

x

^n!1

+ ...+ a

₂

x

²

+ a

₁

x

¹

+ a

₀

Behavior as x gets big?

How many intercepts?

How many turning points?

Graphs of examples

y=x^4-6x^2

-15 -10 -5 0 5 10 15 20 25 30

-4 -3 -2 -1 0 1 2 3 4

f(x)=3x^2+6x-1

-10 0 10 20 30 40 50

-6 -4 -2 0 2 4

What happens if we shift these?

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Graphs of examples

y=x^5-5x^3+4x+1

-3 -2 -1 0 1 2 3 4 5

-3 -2 -1 0 1 2 3

y=x^3-27x

-60 -40 -20 0 20 40 60

0 10 20 30 40 50 60

What happens if we shift these?

Behavior as x gets big?

Behavior as x goes to negative infinity?

Number of intercepts?

Number of turning points?

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Examples

f (x) = x³ ! 2 h(x) = (x)²(x ! 1)

g(x)= (x ! 1)(x ! 2)(x ! 3) j(x) = (x ! 1)(x² + 1)

Behavior as x goes to infinity?

Behavior as x gets to negative infinity?

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In General

f (x) = a

_n

x

ⁿ

+ a

_n_!1

x

ⁿ^!1

+ ...+ a

₂

x

²

+ a

₁

x

¹

+ a

₀

n EVEN:

How many intercepts? Between 0 and n How many turning points? Between 1 and n-1 Continuous?

n Odd:

lim

x!" f (x)= "

lim

x!"# f (x)= "#

lim

x⁺!cf (x)= f (c) = lim

x^"!cf (x)

lim

x!" f (x)=

lim

x!"# f (x)=

a_n > 0 a_n < 0

lim

x!" f (x)= "

lim

x!" f (x)=

a_n > 0 a_n < 0

Finding the roots (or zeros) of a polynomial

• If r is a root (or zero) of the polynomial P(x) this means that P(r) = 0.

• For example,

is a second degree polynomial, and

so r = 4 is a zero of the polynomial as well as (4,0) being an x-intercept of the graph of p(x).

• However for other examples it is not so easy and we do not have a version of the quadratic formula for n>4.

• Luckily there is a Theorem (by Cauchy) to help us out. In the example above it says that any root r of f(x) must satisfy the condition that:

( ) 2 4

p x = x ! x

p(4) =

5 3

( ) 5 4 1

f x = x ! x + x+

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Cauchy’s Theorem approximating roots:

• Given a polynomial

• If r is a root of f(x) then

• Thus we know that all possible x intercepts (roots) are found along the x-axis between ____ and ____. So we know now where to look for the zeros. For example we could set our viewing rectangle on our

calculator to this window and graph the polynomial function.

r < 1+ max{ __ , __ ,..., __ , __ }

f (x) = x

ⁿ

+ a

_n_!1

x

ⁿ^!1

+ ...+ a

₂

x

²

+ a

₁

x

¹

+ a

₀

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Another Example:

• Example: Approximate the real zeros of

• Roots of new polynomial are the same as the roots of P(x).

• We know that all possible x intercepts (roots) are found along the x-axis between ____ and ____. So we set our viewing rectangle on our calculator to this window and graph the polynomial function.

• Plug and chug to see that the root is approximately -3.19 (there is only one root).

3 2

( ) 3 12 9 4 P x = x + x + x+

r < 1+ max{ __ , __ , __ , __ }

Q(x) =

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Rational Function Examples

Graph the following:

f (x)= 1 x

Domain Range

lim

x!" f (x)= lim

x!"# f (x)=

lim

x⁺!2 f (x)= lim

x^"!2 f (x)=

Rational Function Examples

Domain Range lim

x!" f (x)= lim

x!"# f (x)=

g(x)= 1 x + 3 =

lim

x!"

ax+ b cx+ b = lim

x!"

ax

cx+ b+ b cx+ b =

Remark: Limits to infinity in general:

g(1 / 10)=

g(1 / 100)= g(!1 /10) =

g(!1 /100) = How about as x approaches 0?

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Rational Function Examples

f (x)= 1 x ! 2

Domain Range

lim

x!" f (x)= lim

x!"# f (x)=

lim

x⁺!2 f (x)= lim

x^"!2 f (x)=

f (1.9)= f (1.99)

f (2.1)= f (2.01)

How about as x approaches 2?

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Rational Function Examples

Domain Range

lim

x!" f (x)= lim

x!"# f (x)=

lim

x⁺! f (x)= lim

x^"! f (x)=

g(x)= 1

2(x! 1) + 3 =

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Rational Functions

• Definition: A Rational function is a quotient of two polynomials, P(x) and Q(x): R(x)=P(x)/Q(x).

Example: Let P(x) = x + 5 and Q(x) = x – 2 then

R(x)=

Domain:

Range:

Zeros:

x-intercepts:

y-intercepts:

x+ 5 x! 2

Graph of rational function

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Graph of a Rational function:

! Plot points near the value at which the function is undefined. In our case, that would be near x = 2. Plot values such as 1.5, 1.7. 1.9 and 2.1, 2.3, 2.5.

! Determine what happens to the graph of f(x) if x increases or decreases without bound. That is, for x approaching positive infinity or x approaching negative infinity.

! Sketch a graph of a function through these points.

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Rational Functions

• Definition: A Rational function is a quotient of two polynomials, P(x) and Q(x): R(x)=P(x)/Q(x). We will focus on R(x)=

Domain:

Don’t want cx+d = 0. So…

All real numbers except x=-d/c.

The line x=-d/c is the vertical aspmptote Range:

y=a/c is the horizontal asymptote Range: All real numbers except y=a/c.

Zeros: Want ax+b=0

so zero at x=-b/a (if a not zero) x-intercepts: (-b/a,0)

y-intercepts: (0,b/d) lim

x!" f (x)= a / c lim

x!"# f (x)= a / c

lim

x⁺! f (x)= lim

x^"! f (x)=

ax+ b cx+ d

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Example:

f (x) = 3x+ 5 x+ 1

Domain:

Range:

Zeros:

x-intercepts:

y-intercepts:

lim

x!" f (x)= lim

x!"# f (x)=

lim

x⁺! f (x)= lim

x^"! f (x)=