Name: ______________________________ Period:______

Sections 5.2 and 5.2 Guided Notes Lesson will be on October 15, 2013

Instructions: Study sections 5.2 and 5.3 in your book or from my notes on Canvas or my webpage. Fill in the missing information in the notes. On Tuesday, October 15 we will focus on doing problems in class, which will include examples as well as some homework problems. The idea of this activity is to help use class time for hands on practice with the concepts.

Section 5.2 – Properties of Rational Functions

A rational function is a function of the form

Where _____ and _____ are polynomial functions and ____ is not the zero polynomial. The __________ of a _________ function is the _______ of all _________ _______________ except those for which the _________________________ ______ is ________.

Example 1 –

(A) Explain how they got the domain of {x x ≠−5 }.

(B) Explain how they got the domain of {x x≠−2, x≠2 }.

If ____________ is a rational function and if _____ and q have no common factors, then the

___________ function R is said to be in _______________ ___________. For a rational function

_______________ in lowest terms, the ________ _________, if any, of the _____________ in the

domain of R are the ___________________ of the graph of R and so will play a major role in the graph

of R. The real zeros of the ____________________ of R [that is, the numbers x, if any, for which q(x) =

__], although not in the domain of R, also play a major role in the graph of R.

Read through Example 2 – we will go over it again in class.

Read through example 3 – we will go over it again in class.

Asymptotes

In Figure 30, you are given four different graphs that demonstrate asymptotes. In your own words,

explain what happens to the graph R(x) as it approaches the line y = L and explain what asymptote y=L

is.

Which kind of asymptote(s) may a graph intersect? _________________________________

Which kind of asymptote will a graph never intersect?_____________________________

What is the third possible asymptote? ___________________________

What kind of line is this third asymptote? ______________________________

Find the vertical asymptotes in example 4 for parts b and c.

b)

c)

If a rational function is proper, that is, if the ______________ of the numerator is _________ than the

degree of the ______________, then as _________________ or as ______________ the value of R(x)

approaches _____. Consequently, the line ___________ (the ________) is a horizontal _____________

of the graph.

Example 5 – explain why y = 0 is a horizontal asymptote.

If a rational function ______________ is improper, that is, if the ______________ of the

______________ is greater than or ___________ to the _____________ of the ____________________,

we use long division to write the rational function as the sum of a polynomial f(x) (the quotient) plus a

___________ rational function ___________ (r(x) is the remainder). That is we write

Where f(x) is a __________________ and __________ is a proper rational function. Since ___________

is proper, _________________ (which means r(x)/q(x) approaches 0) as _____________ or as

__________. As a result,

The possibilities are listed next. State these possibilities in your own words, go through examples 6 – 8 to help you write them in your own words.

1)

2)

3)

Section 5.3 – The Graph of a Rational Function

Part 1 – Analyze the graph of a Rational Function

List the Step – by – Step Solution and briefly explain how each step affects the graph (Look through examples 1 – 5).

Step 1

Step 2

Step 3

Step 4

Step 5

Step 6

Step 7

Step 8

Constructing the Function from the Graph or problem:

Walk through examples 6 and 7 – Now write your own steps to follow construct a function from a graph or word problem.