This document discusses rational functions and their properties. It defines rational functions as functions defined by a rational expression, which is a polynomial divided by a nonzero polynomial. It explains that the domain of a rational function excludes values where the denominator is zero, as the function is undefined at those points. These values can create holes or vertical asymptotes in the graph. Vertical asymptotes are vertical lines that the graph approaches but does not touch. The document provides steps for simplifying rational expressions and determining holes and vertical asymptotes from the denominator.

1. Algebra 2

2.  How does asymptotes effect a graph?
 How can we come up with the domain and range of
rational functions?
 How do we interpret the graph of rational functions?
 What are vertical asymptotes and how are they used
on a graph?
 Is it possible to simplify rational expressions? If so,
how do we simply them?
 What does it mean if there is a “hole” in our curve?

3. A rational expression consists of a polynomial divided by a nonzero
polynomial. (Examples below)
A rational function is a function defined by a formula that is a rational
expression. The function is in terms of f(x). (Example below)

4. - Domain of a rational function is the set of all real numbers that satisfy
the function except those for which the denominator is zero.
- Find the domain by determining when the denominator is zero. When
the denominator equals zero the function and graph will be undefined.
- We want to find an (x) that makes the function undefined creating a
“hole” in the functions graph.
- Therefore, set the denominator equal to zero and solve for x. Note: There
may be more than one number that sets the denominator equal to zero.

6. Expressions
- A rational expression is simplified if its numerator and denominator have no
common factors other than 1 or -1.
Steps to simplify rational expressions:
1.) Factor the numerator and the denominator completely.
2.) Divide both the numerator and the denominator by any common factors.
(only divide out, or cancel factors common to the numerator and
denominator).

7. Vertical Asymptotes
-The denominator also gives an (x value) where there is a
vertical asymptote
Vertical asymptote is a vertical line that the graph of a
function approaches, BUT DOES NOT TOUCH.
a.) a graph may have zero, one, or several
veritcal asymptotes
b.) no point from the function can land on an
asymptote, however, it can be extremely close.
c.) MUST check the x-value that makes the
denominator zero by putting it into the function and
seeing if it is a vertical asymptote or hole by the
solution it gives.

8.  Does every rational function have values to exclude? (x
values where the denominator equals zero)
 How can you tell if an x-value (from the denominator)
is going to be a “hole” or “vertical asymptote” on a
graph?
 Can you factor the denominator and set each factors
equal to zero to solve for x?

9.  Does every rational function have values to exclude? (x values
where the denominator equals zero)
No, (x squared plus 1) has no real number values that cause
the denominator to equal zero.
 How can you tell if an x-value (from the denominator) is going to
be a “hole” or “vertical asymptote” on a graph?
After you solve for x, substitute the x value into the equation
and solve. If the solution is any number besides zero then it
is a “hole”. If the solution is zero then there is a vertical
asymptote at that x value.
 Can you factor the denominator and set each factors equal to
zero to solve for x?
Yes, but ONLY if you are looking for the domain, hole, or
vertical asymptote.

10. The graph of the function
is given in green.
(if you take random x-
values and substitute it in
the equation the solution
will give a point (x,y) that
will be along one of the
green curves).
The vertical asymptote is
where x = -2 (from the
denominator).
When x = -2, the function
is undefined, therefore,
there can not be an x-value
that equals -2.