2. 8.2 Rational Functions and Their Graphs
Objectives:
Identify and evaluate rational functions.
Graph a rational function, find its domain, write equations for its asymptotes, and identify any holes in its graph.

3. A rational expression is the quotient of two polynomials.
A rational function is a function defined
by a rational expression.
Determine whether the function is a rational function: No
1.) f(x) =
2.)
Yes

4. The rational function f(x) = 1/x is
undefined when x=0.
In general, the domain of a rational
function is the set of all real
numbers except those numbers which make the denominator equal to zero.

5. Example 2

7. Rational functions can have horizontal and vertical asymptotes.

8. Real numbers for which a rational function is not defined are called excluded values. At an excluded value, a rational function may have a vertical asymptote. If x – a is a factor of the denominator of a rational function but not a factor of its numerator, then x = a is a vertical asymptote of the graph of the function.

9. Example 3

12. Example 4

14. Example 5

15. Vertical stretch by a factor of 4
Translation 1 unit up
Translation 2 units to the right

16. The graph of a rational function may have a
hole in it. For example, can be
written as because x - 3 is a
factor of both the numerator and the
denominator, the graph of f has a hole
when x = 3.
hole when x = 3
hole when x = 3

17. If x – b is a factor of the numerator and
the denominator of a rational function,
then there is a hole in the graph of a
function when x = b, unless x = b is a
vertical asymptote.

18. Identify all asymptotes and holes in the graph.
Example 6. Let:
Identify all asymptotes and holes in the graph.
Because x-1 is a factor of both the
numerator and denominator, the
graph has a hole when x = 1
Because x+2 is a factor of only the
denominator, there is a vertical
asymptote. x= -1
Because the degree of the numerator
equals the degree of the denominator
there is a horizontal asymptote as y= -1.

21. Homework
Integrated Algebra II- Section 8.2 Level A
Academic Algebra II- Section 8.2 Level B