1. Factor completely and simplify. State the domain.
List all possible rational zeros of
f (x) = 2x^ 4 – x ^3 – 3x ^2 – 31x – 15
Find all the rational zeros of
g (x) = x^ 4 – x^ 3 + 2x^ 2 – 4x – 8

2. Lesson 2.5: Rational Function

3. Lesson 2.5 Rational Function
A rational function can be written in the form , where N(x) and D(x) are polynomials and D(x)
Example :

5. Horizontal and Vertical Asymptotes
The domain of the a rational function are all real number except x-values that make the denominator D(x) = 0
Horizontal and Vertical Asymptotes
1. The line x = a is a vertical asymptote of the
graph of f if as
either from left or right.
2. The line y = b is a horizontal asymptote of the
graph of f if

7. Asymptotes of Rational Function
Let
where N(x) and D(x) have no common factors:
1. There are vertical asymptotes at the Zero of D(x)
2. The graph has one or no horizontal asymptote determined by comparing the degree of N(x) and D(x)
If n< m , y = 0 is the horizontal asymptote
If n = m , the line is a horizontal
asymptote
If n > m , no horizontal asymptote

8. Guidelines for Analyzing Graphs of Rational Functions
1.Simplify if possible 2. Plot y – intercept(if any) by evaluating f(0) ; plot hole by evaluating simplified f. 3. Find zeros of numerator to find x – intercept 4. Find zeros of denominator to find vertical asymptotes. 5. Find and sketch any horizontal asymptotes. (Use the rule for H.A.) 6. Mini graphs (plot points between x-int and V.A.) 7. Use smooth curves to complete the graphs.

9. Example : Find the intercepts, vertical and horizontal asymptotes
Example : Find the intercepts, vertical and horizontal asymptotes. Then sketch the graph of each function. (no common factor)

10. With Common Factor/s
Find all asymptotes and sketch the graph.

11. Examples:

12. Slant Asymptote
If the degree of the numerator is exactly one more than the degree of the denominator, the graph of the function has a slant/oblique asymptote. (Use long division) Example: Sketch the graph of a. b.