1. Rational Functions and Asymptotes
Section 2.6
Rational Functions and Asymptotes

2. 2.6 Introduction to Rational Functions
A rational function is a function of the form f(x) = N(x)/D(x), where N and
D are both polynomials and D(x) The domain of f is all real x’s except x values that give 0 in the denominator. N(x) and D(x) should have no common factors.
Example 1
Domain is all reals except x=0
Plug in x value to find y value.
This table shows x approaching 0(the excluded x value) from the left. x -1
-1/2
-1/10
-1/100
-1/1000
f(x) -2
-10
-100
-1000
This table shows x approaching 0 from the right. x
1/1000
1/100
1/10
1/2 1
f(x)
1000
100 10 2

3. x f(x) x f(x) -1 -1/2 -1/10 -1/100 -1/1000 -2 -10 -100 -1000 1/1000
f(x) -2
-10
-100
-1000 x
1/1000
1/100
1/10
1/2 1
f(x)
1000
100 10 2
Plot the sets of ordered pairs and this is the graph that you get.
Note that as x approaches 0 from the left, f(x) decreases without bound. In contrast, as x approaches 0 from the right, f(x) increases without bound. x
f(x)
Remember that the domain is all reals except x=0.
What do you notice about the line x=0 and the graph of f ???
The graph never touches the line. This line is a vertical asymptote.

4. Vertical and Horizontal Asymptotes
1. The line x = a is a vertical asymptote of the graph of f if f(x) as x a, either from the right or left.
2. The line y = b is a horizontal asymptote of the graph of f if f(x) b as x

5. Asymptotes of Rational Functions Rules:
where N(x) and D(x) have no common factors.
The graph of f has vertical asymptotes at the zeros of D(x).
The graph of f has at most one horizontal asymptote determined by comparing the degrees of N(x) and D(x).
a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote.
b. If n=m, the line y=an/bm is a horizontal asymptote.
c. If n>m , the graph of f has no horizontal asymptote.

6. Asymptotes of Rational Functions Rules:
Slant Asymptotes
Only occur when the degree of the top is 1 more than the degree of the bottom.
The S.A. is derived by dividing the top by the bottom (long division or synthetic) and ignoring the remainder.

7. a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote.
b. If n=m, the line y=an/bm is a horizontal asymptote.
c. If n>m , the graph of f has no horizontal asymptote.
Ex Find the Horizontal Asymptotes for each of the following functions.
n<m therefore the horizontal asymptote is y=0.

8. a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote.
b. If n=m, the line y=an/bm is a horizontal asymptote.
c. If n>m , the graph of f has no horizontal asymptote.
Ex Find the Horizontal Asymptotes for each of the
following functions.
n=m therefore, y=2/3 is the horizontal asymptote.

9. a. If n<m, the line y=0 ( the x-axis) is a horizontal asymptote.
b. If n=m, the line y=an/bm is a horizontal asymptote.
c. If n>m , the graph of f has no horizontal asymptote.
Ex. 2 Find the Horizontal Asymptotes for each of the following functions.
n>m, therefore there is no horizontal asymptote.
Although this graph does not have a horizontal asymptote it does have a slant or oblique asymptote – the line y=2/3 x.

10. Ex. 3 For the function f, find a) the domain of f, b) the vertical asymptotes of f, and c) the horizontal asymptote of f.
a) Set denominator =0 and solve.
b) The graph of f has vertical asymptotes at the zeros of D(x). Therefore the vertical asymptote of f is
c) If n=m, the line
y=an/bm is a horizontal
asymptote. Therefore, the
horizontal asymptote is

12. Ex. 4 A Graph with Two Horizontal Asymptotes A function that is not rational can have two horizontal asymptotes-one to the left and one to the right. For instance, the graph of
HA y = -1
HA y = 1

13. Find the following if possible: domain, vertical asymptote, horizontal asymptote, slant asymptote.

14. Find the following if possible: domain, vertical asymptote, horizontal asymptote, slant asymptote.

15. Ultraviolent Radiation
For a person with sensitive skin, the amount of time T (in hours) the person can be exposed to the sun with mininal burning can be modeled by
where s is the Sunsor Scale Reading. The Sunsor Scale is based on the level of intensity of UVB rays.

16. Homework
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