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Warm UP!. Factor the following:. Unit 7: Rational Functions. LG 7-1: Characteristics & Graphs of Rationals (quiz 11/6) LG 7-2: Inverses of Rational Functions (quiz 11/8) LG 7-3: Solving Rational Equations & Inequalities (quiz 11/13) TEST 11/15. Rational Functions. General Equation:

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Warm up
Warm UP!

Factor the following:


Unit 7 rational functions

Unit 7: Rational Functions

LG 7-1: Characteristics & Graphs of Rationals

(quiz 11/6)

LG 7-2: Inverses of Rational Functions

(quiz 11/8)

LG 7-3: Solving Rational Equations & Inequalities

(quiz 11/13)

TEST 11/15


Rational functions
Rational Functions

  • General Equation:

    • S and T are polynomial functions

  • Verbally: f(x) is a rational function of x.

  • Features: A rational function has a discontinuities- asymptotes and/or holes.


The parent rational function is:

  • The shape is made by the behavior of a function as it approaches asymptotes.

  • ALL rational functions will look similar to this parent graph as they are all part of the same family.

An example of a rational function is:


How to find the characteristics of rational functions

How to find the Characteristics of Rational Functions

Domain & Range

Intercepts

Discontinuities - Vertical Asymptotes, Horizontal Asymptotes, Slant/Oblique Asymptotes, and Holes


All rational functions have asymptotes in their graphs
All rational functions have asymptotes in their graphs.


X-intercepts

Where the function crosses the x-axis.

A function can have none, one, or multiples

To find the x-int of a Rational Function, set the numerator equal to zero and solve for x.


Practice

Find all x-intercepts of each function.


y-intercepts

Where the function crosses the y-axis

A function can have NO MORE THAN 1!

To find the y-int of Rational Functions, substitute 0 for x.


Practice

Find all y-intercepts of each function.


Vertical asymptotes
Vertical Asymptotes

  • A vertical asymptote is an invisible line that the graph will NEVER cross.

  • The function is undefined at a VA.

  • You can find the VA by setting the denominator equal to zero and solving for x.


Domain of a rational function
Domain of a Rational Function

  • The domain of a rational function is all real numbers excluding the discontinuities (the vertical asymptote) and the x-value of the hole (more tomorrow!)


When you have one vertical asymptote
When you have one vertical asymptote…

  • Your graph is separated into 2 sections…

    • To the left of the asymptote

    • To the right of the asymptote


When you have two vertical asymptotes
When you have two vertical asymptotes…

  • Your graph is separated into 3 sections…

    • To the left of both asymptotes

    • In between the asymptotes

    • To the right of both asymptotes


Practice

Find the Vertical Asymptotes & Domain:


Horizontal asymptotes
Horizontal Asymptotes

  • A horizontal asymptote is an invisible line that the graph will SOMETIMES cross.

  • The function is undefined at a HA.

  • You can find the HA by using the same rules we discussed for limits – use the DEGREES 


Range of a rational function
Range of a Rational Function

  • The range of a rational function is all real numbers excluding the horizontal asymptote (and the y-coordinate of a hole – more tomorrow!)


Practice

Find the Horizontal Asymptotes & Range:


How do you find asymptotes in rational functions
How do you find asymptotes in rational functions?

Horizontal Asymptotes

Vertical Asymptotes

Compare the degrees of the numerator and the denominator

1. Set denominator equal to zero.

1. If bigger on BOTTOM then there is a HA at y = 0.

2. Solve for x

2. If n = m, then there is a HA at y = ___ (find by divide the leading coefficients).

3. If bigger on TOP, then there is no horizontal asymptotes (this means there could be a SLANT/OBLIQUE asymptote which we will discuss tomorrow!


Find all asymptotes of

Vertical:

x = -1

and x = 2

Horizontal:

Degree of top = 1

Degree of bottom = 2

BIGGER on BOTTOM

y = 0


Find all asymptotes of

x = 0

Vertical:

Horizontal:

Degree of numerator = 1

Degree of denominator = 1

EQUAL


Find all asymptotes of

Vertical:

x - 1 = 0

x = 1

Degree of numerator = 2

Horizontal:

Degree of denominator = 1

Bigger on TOP

No horizontal asymptote


For example:

Find the following characteristics for the rational function below:

Domain:

Range:

x-intercepts:

y-intercepts:

Horizontal asymptote:

Vertical asymptotes:


Find all asymptotes of

Ex. 4

and graph

Vertical:

x + 1 = 0

Set denominator equal to zero

x = -1

Horizontal:

n > m by exactly one

n = 2

m = 1

No horizontal asymptote

Compare degrees

Slant:

Use long division




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