Warm up
This presentation is the property of its rightful owner.
Sponsored Links
1 / 26

Warm UP! PowerPoint PPT Presentation


  • 121 Views
  • Uploaded on
  • Presentation posted in: General

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:

Download Presentation

Warm UP!

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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.


Warm up

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.


Warm up

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.


Warm up

Practice

Find all x-intercepts of each function.


Warm up

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.


Warm up

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


Warm up

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!)


Warm up

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!


Warm up

Find all asymptotes of

Vertical:

x = -1

and x = 2

Horizontal:

Degree of top = 1

Degree of bottom = 2

BIGGER on BOTTOM

y = 0


Warm up

Find all asymptotes of

x = 0

Vertical:

Horizontal:

Degree of numerator = 1

Degree of denominator = 1

EQUAL


Warm up

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


Warm up

For example:

Find the following characteristics for the rational function below:

Domain:

Range:

x-intercepts:

y-intercepts:

Horizontal asymptote:

Vertical asymptotes:


Warm up

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


What are your questions

What are your questions?


Characteristics of rational functions

Characteristics of Rational Functions

Vertical Asymptotes


  • Login