2.6 Rational Functions - Kenton Elementary School Rat Func Beamer.pdf · The Domain of Rational Functions In general, thedomainof a rational function of x includes all real numbers

2.6 – Rational Functions

Pre-Calculus

Mr. Niedert

Pre-Calculus 2.6 – Rational Functions Mr. Niedert 1 / 20


1 Rational Functions

2 Horizontal and Vertical Asymptotes

3 Oblique/Slant Asymptotes












Definition of a Rational Function

A rational function can be written in the form

f (x) =N(x)

D(x)

where N(x) and D(x) are each polynomials and D(x) is not the zeropolynomial.


The Domain of Rational Functions

In general, the domain of a rational function of x includes all realnumbers except x-values that make the denominator zero.


Finding the Domain of a Rational Function

Example

Find the domain of f (x) =3x

x − 1. Express your answer in interval

notation. Then describe the behavior of the function f near any excludedx-values.


Finding the Domain of a Rational Function

Practice

Find the domain of f (x) =5

x2 − 4x + 3. Express your answer in interval

notation. Then describe the behavior of the function f near any excludedx-values.


Exit Slip

Exit Slip

Find the domain of each rational function. Express your answer in intervalnotation.

a5x

x + 2

b3x2

1 + 3x

c8

x2 − 10x + 24

dx2 + x − 2

x2 + 4


Vertical Asymptotes

Yesterday, we looked at the polynomial y =3x

x − 1. Below is a graph

of the equation.

Notice that f (x) decreases without bound as x approaches 1 from theleft. In addition, f (x) increases without bound as x approaches 1from the right. This can be denoted as follows.

f (x)→ −∞ as x → 1− f (x)→∞ as x → 1+

This then means that x = 1 is a vertical asymptote of the graph of

y =3x

x − 1.


Vertical Asymptotes



of the equation.

Notice that f (x) decreases without bound as x approaches 1 from theleft. In addition, f (x) increases without bound as x approaches 1from the right.

This can be denoted as follows.

f (x)→ −∞ as x → 1− f (x)→∞ as x → 1+


y =3x

x − 1.


Vertical Asymptotes



of the equation.


f (x)→ −∞ as x → 1− f (x)→∞ as x → 1+


y =3x

x − 1.


Vertical Asymptotes



of the equation.


f (x)→ −∞ as x → 1− f (x)→∞ as x → 1+


y =3x

x − 1.


Horizontal Asymptotes

Again, let’s continue to consider the polynomial y =3x

x − 1.

Notice that f (x) approaches 3 as x decreases without bound andf (x) approaches 3 as x increases without bound. This can be denotedas follows.

f (x)→ 3 as x → −∞ f (x)→ 3 as x →∞This then means that y = 3 is a horizontal asymptote of the graph of

y =3x

x − 1.




x − 1.

Notice that f (x) approaches 3 as x decreases without bound andf (x) approaches 3 as x increases without bound.

This can be denotedas follows.


y =3x

x − 1.




x − 1.


f (x)→ 3 as x → −∞ f (x)→ 3 as x →∞

This then means that y = 3 is a horizontal asymptote of the graph of

y =3x

x − 1.




x − 1.



y =3x

x − 1.


Asymptotes of a Rational Function


Let f be the rational function given by

f (x) =N(x)

D(x)

where N(x) and D(x) are each polynomials and have no common factors.

The graph of f has vertical asymptotes at the zeros of D(x).

The graph of f has one or zero horizontal asymptotes, which we candetermine by comparing the degrees of N(x) and D(x). For the sakeof explanation, let n represent the degree of N(x) and d represent thedegree of D(x).

I n < d =⇒ the graph of f has the line y = 0 (the x-axis) as ahorizontal asymptote.

I n = d =⇒ the graph of f has the line y = anbd

where an and bd are the

leading coefficients of N(x) and D(x), respectively.I n > d =⇒ the graph of f has no horizontal asymptote.





f (x) =N(x)

D(x)












f (x) =N(x)

D(x)












f (x) =N(x)

D(x)







leading coefficients of N(x) and D(x), respectively.

I n > d =⇒ the graph of f has no horizontal asymptote.





f (x) =N(x)

D(x)









Finding Horizontal and Vertical Asymptotes

Example

Find all horizontal and vertical asymptotes of the graph of

f (x) =2x2 − 7x + 3

x2 − 5x + 6.


Finding Horizontal and Vertical Asymptotes

Practice

Find all horizontal and vertical asymptotes of the graph of each rationalfunction below.

a f (x) =2x2

x2 − 1

b f (x) =5

x − 2

c f (x) =x2 + x − 2

x2 − x − 6

d f (x) =x2

x − 4


Exit Slip

Exit Slip

Find all horizontal and vertical asymptotes of the graph of each rationalfunction below.

a f (x) =x3

x2 − 4

b f (x) =x − 6

x2 − 8x + 12

c f (x) =x2 − 6x + 9

x2 − 7x + 10


Rational Functions Assignment 1

Due Next Class: pg. 193 #5-16


Asymptotes

Discovery

Look at each of the graphs below. Just from the graphs, estimate wherethere might be asymptotes.

Graph 1 Graph 2 Graph 3


Oblique/Slant Asymptotes

As we saw on the previous slide, Graph 3 has a oblique (or slant)asymptote.

Graph 3 is the graph of the function

f (x) =x2 − x

x + 1.

It has an oblique/slant asymptote because the degree of thenumerator (n) is exactly one degree greater than the degree of thedenominator (d).


Oblique/Slant Asymptotes

As we saw on the previous slide, Graph 3 has a oblique (or slant)asymptote.

Graph 3 is the graph of the function

f (x) =x2 − x

x + 1.

It has an oblique/slant asymptote because the degree of thenumerator (n) is exactly one degree greater than the degree of thedenominator (d).


A Rational Function with a Slant Asymptote

Example

Find all of the asymptotes (vertical, horizontal, and/or oblique/slant) of

the function f (x) =x2 − x − 2

x − 1.


A Rational Function with a Slant Asymptote

Practice

Find all of the asymptotes of the function f (x) =3x2 + 1

x.


Exit Slip

Exit Slip

The asymptotes of f (x) =x2 − x

x + 1(from Graph 3 earlier) were at x = −1

and y = x − 2. Describe and/or show why this is the case.


Rational Functions Assignment 2

Due Next Class: Rational Functions Assignment 2 Worksheet

This is the end of the section, so I should have Assignment 1 (pg. 193#5-16) by the next class as well.


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2.6 Rational Functions - Kenton Elementary School Rat Func Beamer.pdf · The Domain of Rational Functions In general, thedomainof a rational function of x includes all real numbers