Download presentation

1
**Section 5.2 Properties of Rational Functions**

Objectives Find the Domain of a Rational Function Determine the Vertical Asymptotes of a Rational Function Determine the Horizontal or Oblique Asymptotes of a Rational Function

2
**A rational function is a function of the form**

where p and q are polynomial functions and q is not the zero polynomial. The domain consists of all real numbers except those for which the denominator q is 0.

3
**Find the domain of the following rational functions.**

All real numbers x except -6 and -2. All real numbers x except -4 and 4. All Real Numbers

4
**Recall that the graph of is**

(1,1) (-1,-1)

5
**Graph the function using transformations**

(1,1) (-1,-1) (3,1) (1,-1) (2,0) (3,2) (1,0) (2,0) (0,1)

6
If, as x or as x , the values of R(x) approach some fixed number L, then the line y = L is a horizontal asymptote of the graph of R. If, as x approaches some number c, the values |R(x)| , then the line x = c is a vertical asymptote of the graph of R. In the previous example, there was a vertical asymptote at x = 2 and a horizontal asymptote at y = 1. (3,2) (1,0) (2,0) (0,1)

7
**Examples of Horizontal Asymptotes**

y = L y = R(x) y x y = L y = R(x) y x

8
**Examples of Vertical Asymptotes:**

x = c y x = c y x x

9
**Theorem: Locating Vertical Asymptotes **

If an asymptote is neither horizontal nor vertical it is called oblique. y x Theorem: Locating Vertical Asymptotes A rational function R(x) = p(x) / q(x), in lowest terms, will have a vertical asymptote x = r, if x - r is a factor of the denominator q.

10
**Vertical asymptotes: x = -1 and x = 1**

Example: Find the vertical asymptotes, if any, of the graph of each rational function. Vertical asymptotes: x = -1 and x = 1 No vertical asymptotes Vertical asymptote: x = -4

11
**Consider the rational function**

in which the degree of the numerator is n and the degree of the denominator is m. 1. If n < m, then y = 0 is a horizontal asymptote of the graph of R. 2. If n = m, then y = an / bm is a horizontal asymptote of the graph of R. 3. If n = m + 1, then y = ax + b is an oblique asymptote of the graph of R. Found using long division. 4. If n > m + 1, the graph of R has neither a horizontal nor oblique asymptote. End behavior found using long division.

12
**Horizontal asymptote: y = 0 Horizontal asymptote: y = 2/3**

Example: Find the horizontal or oblique asymptotes, if any, of the graph of Horizontal asymptote: y = 0 Horizontal asymptote: y = 2/3

13
**Oblique asymptote: y = x + 6**

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google