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Warm UP!

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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

- 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

Domain & Range

Intercepts

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

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.

- 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.

- 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!)

- Your graph is separated into 2 sections…
- To the left of the asymptote
- To the right of the asymptote

- 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:

- 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

- 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:

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

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

Characteristics of Rational Functions

Vertical Asymptotes