1. Rational Functions Part One
Mathematics Precalculus with trigonometry
Fall 2016

2. Announcements and Questions
* Laboratory One due 1 November
* Exams, other papers, and grade sheets
on Wednesday.
* Do you have any questions?

3. What are Rational Functions?
Functions expressed as the ratio of functions;
often polynomial functions.
N for numerator
D for denominator
What is the Domain of a Rational Function?
All x-values except those that make the denominator zero or make D(x) undefined.

4. This graph looks a little strange…….
Example 1:
Pull out your calculators and graph the function.
This graph looks a little strange…….

5. What we see …..
What we should think…
These graphs are the same…it just looks like the graph disappears…in actuality, the graph keeps getting closer and closer and closer and closer and closer and closer to both the x and y axes. The calculator has some difficulty showing this for us.

6. Example 1: (Continued) What is the domain? For a Rational Function
the domain are all values for which the D(x) ≠ 0 or where D(x) is not defined.
Since D(x) = x, we have
that x ≠ 0. So the
Domain is R - { 0 }
To find Zeros we set the N(x) = 0. Since this is impossible
there are no Zeros.
Since f(0) is undefined. There is no y-intercept either.

7. Example 1: (Continued)
The search for Horizontal Asymptotes
What happens to the curve
as x → ∞ ?
f(x) → 0
What happens to the curve
as x → − ∞ ?
f(x) → 0
So we say there is a horizontal asymptote (HA) at y = 0.
Horizontal Asymptotes describe the behavior of the
function at the ends!

8. Example 1: (Continued)
The search for Horizontal Asymptotes
with the TI-89 Calculator
Home => F3: Calculus => 3:limit(
You complete the function call by
adding parameters- function,
variable and “infinity symbol.”
So our command looks like
this
limit(y1(x), x, ∞)

9. We have the Horizontal Asymptote!!
Example 1: (Continued)
The search for Horizontal Asymptotes
with the TI-89 Calculator
Enter this command for x → ∞.
limit(y1(x), x, ∞) 
lim x → ∞ f(x) = 0
Enter this command for x → − ∞
limit(y1(x), x, − ∞) 
lim x → − ∞ f(x) = 0
We have the Horizontal Asymptote!!

10. Example 1: (Continued)
The search for Vertical Asymptotes
The search for vertical asymptotes is related to our discovering the domain.
Vertical Asymptotes exist where the denominator would be zero
They are shown as
Vertical Dashed Lines
They are NOT part of the graph
There can be more than one!

11. Example 1: (Continued)
The search for Vertical Asymptotes
To find them, set the denominator function equal to zero and solve for “x”
So for our function f, we have
Vertical Asymptote at x = 0!

12. Vertical Asymptotes with the TI-89 Calculator
Example 1: (Continued)
Vertical Asymptotes with the
TI-89 Calculator
Home => F3: Calculus => 3:limit(
Enter this command for x → 0 −
limit(y1(x), x, 0, - 1 ) 
lim x → 0- f(x) = − ∞
Enter this command x → 0 +
limit(y1(x), x, 0, 1 ) 
lim x → 0+ f(x) = ∞
You can verify the calculator work by looking at the graph!

13. From the calculator we have the following graph The Analysis of f
Problem 1 :
From the calculator we have
the following graph
The Analysis of f
Domain : R − { 0 }
Vertical Asymptote at x = 0
NOTE: It is only when very far away from the origin that the graph approaches an asymptote…
Horizontal Asymptote at y = 0
There are No Zeros and
no y-intercept
Range (0, ∞ )

14. From the calculator we have the following graph
Problem 2 :
The Analysis of f
From the calculator we have
the following graph
Domain : R − { − 1 }
Vertical Asymptote at x = − 1
Using the TI-89 Calculator we
Have that the Horizontal Asymptote
is at y = 2
To find the Zeros set the
Numerator function to 0.
Solve 2x + 1 = 0. We have
a zero at (− ½ , 0)
To find the y-intercept find f(0).
So the y-intercept is ( 0, 1 )
Range R − { 2 }

15. Hints for successful graphing Rational Functions
0. Use your calculator to sketch the graph
1. Set x = 0. Plot the y-intercepts.
2. Find the zeros of function by setting N(x) = 0. Plot the x-intercepts.
Find the zeros of the denominator by setting D(x) = 0. Sketch the VA Use your CALCULATOR to help you.
Find and sketch any HA by using the limit function of your CALCULATOR
5. Plot any points as needed…..
Draw nice smooth curves and then say
“Job Well Done!!”

16. Your Homework will be posted by 3 PM