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### Unit 7: Rational Functions

### How to find the Characteristics of Rational Functions

### Characteristics of Rational Functions

Warm UP!

Factor the following:

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:
- 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:

Domain & Range

Intercepts

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

All rational functions have asymptotes in their graphs.

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.

Find all x-intercepts of each function.

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.

Find all y-intercepts of each function.

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

- 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…

- 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…

- Your graph is separated into 3 sections…
- To the left of both asymptotes
- In between the asymptotes
- To the right of both asymptotes

Find the Vertical Asymptotes & Domain:

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

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

Find the Horizontal Asymptotes & Range:

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!

Vertical:

x = -1

and x = 2

Horizontal:

Degree of top = 1

Degree of bottom = 2

BIGGER on BOTTOM

y = 0

x = 0

Vertical:

Horizontal:

Degree of numerator = 1

Degree of denominator = 1

EQUAL

Vertical:

x - 1 = 0

x = 1

Degree of numerator = 2

Horizontal:

Degree of denominator = 1

Bigger on TOP

No horizontal asymptote

Find the following characteristics for the rational function below:

Domain:

Range:

x-intercepts:

y-intercepts:

Horizontal asymptote:

Vertical asymptotes:

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

Vertical Asymptotes

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