1. 3.5 Rational Functions
An introduction

2. Objectives: Identify key characteristics of rational functions.
Determine the domain and range of a rational function.
Identify all asymptotes of a rational function.
Locate removable discontinuities of a rational function.

3. A function is CONTINUOUS if you can draw the graph
without lifting your pencil.
A POINT OF DISCONTINUITY occurs when
there is a break in the graph.
Note the break in the graph
when x=3. Why?

4. Look at the equation of the graph.
Where is this equation undefined?
We can factor the numerator and reduce
the fraction to determine that the graph
will be a straight line; however, the
undefined point remains, so there is a
point of discontinuity here.

5. There are three basic kinds of discontinuity:
point, jump, and infinite.
The greatest integer function is an example of a
jump discontinuity.
Tangent, Cotangent, secant and cosecant
functions are all examples of infinite discontinuities.
The previous function was an example of a
point of discontinuity.

6. A rational function is a quotient of two polynomials.
The graphs of rational functions frequently
display infinite and point discontinuities.
Rational functions have vertical asymptotes and
may have horizontal asymptotes as well.
Recall that a vertical asymptote occurs when
there is a value for which the function is undefined.
Since a rational function is a quotient of two,
polynomials, there will almost always be at least
one value for which the entire function is undefined.

7. Let’s look at the parent function:
If x = 0, then the entire function is undefined.
Thus, there is a vertical asymptote at x=0.
Looking at the graph, you can see that the value of the
function , as the values of x from the positive
side; and the value of the function , as the values
of x from the negative side. These are the limits of
the function and are written as:

8. Domain To find the domain of a rational function,
The domain is then limited to:
To find the domain of a rational function,
set the denominator equal to zero.
The domain will always be all real
numbers except those values found by
solving this equation.

9. Determine the domain of these
rational functions:

10. Recall that a vertical asymptote occurs when
there is a value for which the function is
undefined. This means, if there are no
common factors, anywhere the denominator
equals zero.

11. Remember that asymptotes are lines
Remember that asymptotes are lines. When you label a vertical asymptote, you must write the equation of the vertical line.
Just make x equal everything it couldn’t in the domain.
State the vertical asymptotes:

12. Let’s say x is any positive number. As that value increases,
the value of the entire function decreases; but, it will
never become zero or negative.
So this part of the graph will never
cross the x-axis. We express this
using limit notation as:
What if x is a negative number? As that value decreases, the value of the entire function increases; but, it will never become zero or positive.
So this part of the graph will never
cross the x-axis, either and:

13. Thus, the line is a horizontal asymptote.
As x ∞, f(x) , and as x ∞, f(x)
Given: is a polynomial of degree n , is a
polynomial of degree m , and , 3 possible
conditions determine
a horizontal asymptote:
If n<m, then is a horizontal asymptote.
If n>m, then there is NO horizontal asymptote.
If n=m, then is a horizontal asymptote, where c
is the quotient of the leading coefficients.

14. BOB0 BOTN EATS DC

15. B O B B O T N E A T S - D C I G E R N O T M Z E R O I G E R N O P O N E X P O N E T S R E H E A M E I V D E O E F I C N T S

16. Find the horizontal asymptote:
Exponents are the same; divide the coefficients
Bigger on Top; None
Bigger on Bottom; y=0

17. Identify the vertical and/or horizontal asymptotes for
each function.

18. Suppose that you were asked to graph:
1st, determine where the graph is undefined.
(Set the denominator to zero and solve for the variable.)
There is a vertical asymptote here.
Draw a dotted line at:
2nd , find the x-intercept by setting the numerator = to 0
and solving for the variable.
So, the graph crosses the x-axis at

19. 3rd , find the y-intercept by letting x=0 and solving for y.
So, the graph crosses the y-axis at
4th , find the horizontal asymptote. Thus, Use Bob0 Botn
Eats DC….the largest degreed term on the top divided
by largest degreed term on the bottom: 𝟑𝒙 𝒙
This is Eats DC , so divide the coefficients: 𝟑 𝟏
The horizontal asymptote is:

20. Now, put all the information together and sketch
the graph:

21. Graph:
1st, find the vertical asymptote.
2nd , find the x-intercept .
3rd , find the y-intercept.
4th , find the horizontal asymptote.
5th , make a sign chart: (next slide).

22. Sign Charts:
𝟒𝒙+𝟑
𝒙−𝟓
Answer
− 𝟑 𝟒 𝟓
Write the domain and range:

23. 1st, factor the entire equation:
Graph:
1st, factor the entire equation:
Then find the vertical asymptotes:
2nd , find the x-intercepts:
3rd , find the y-intercept:
4th , find the horizontal asymptote:
5th , Make a sign chart (next slide)

24. 𝒙+𝟏
𝒙−𝟐
𝒙−𝟑
𝒙+𝟐
Answer
Write the domain and range:

25. Graph:
Notice that in this function, the degree of the
numeratoris larger than the denominator. Using
Bob0 BotN – we have bigger on top so there is no
horizontal asymptote. However, if n is exactly one more
than m, the rational function will have a slant or Oblique
asymptote.
To find the slant asymptote, divide the numerator by the
denominator:
The result is Notice that as the values of x
increase, the fractional part decreases (goes to 0), so the
function approaches the line Thus the line
is a slant asymptote.

26. Graph:
1st, find the vertical asymptote.
2nd , find the x-intercepts: and
3rd , find the y-intercept:
4th , find the horizontal asymptote OR
Find the slant asymptote:
5th , Make a sign chart. (Next Slide)
6th , Sketch the graph. (Next Slide)
7th, Identify Domain and Range. (Next Slide)

27. Write the domain and range:𝒙
𝟐𝒙−𝟑
𝒙+𝟏
Answer
Write the domain and range:

29. Write a function with the following characteristics:
19. A vertical asymptote at
A horizontal asymptote at
An x-intercept at

30. 20. A vertical asymptote at
An oblique asymptote at

31. How to Proceed:
Please make sure you interact with the online lesson now. Interact means that you do not just watch, rather you participate with the slides when given that opportunity and you take notes as needed.
Then, you need to attempt to do the entire check up without peaking at the answers until you are finished – at which time you should check your work against the answers and then try to discern why you missed a problem – if you did!
Then you should review your notes, your check up and attend CC or PASS and then, when you are certain you will do well on the quiz…then and only then you should sit down in a testing situation and complete the quiz.

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