# Notes for Rational Functions

Topics: Graphing Rational Functions , Finding Vertical and

Horizontal Asymptotes, Applications of Rational Functions

This video lesson has lots of material that you will omit. You may want to go
through this handout

first, the book second, then look at the video if necessary.

Recall: A** rational number **can be expressed as the quotient of two
integers
.

A **rational function** is the quotient of two polynomials
, where q(x) does not equal 0.

*Remember when entering fractions like the 2^{nd}, 3^{rd} and
4^{th} examples into Y_{1}, you must use ( )

around each numerator and/or denominator since they have a sum or difference in
each part. I have

set the WINDOWS at either [-8, 8, 1] x [-8, 8, 1] or [-5, 5 1] x [-5, 5, 1]

* The graphs above are in from a MODE that is CONNECTED, where the calculator is
doing its

best to draw you a continuous graph. The MODE that is DOT will give you a graph
like:

Choose whichever you prefer , as long as you
realize the difference in the looks of the two . More about why they are different later … |

I. Graphing Rational Functions (p. 350 – 353)

These graphs are not your every-day graphs. They have
curves and bends that are not parabolic,

cubic, or any like we’ve seen.

They have some common characteristics that will help to sketch them:

1. The **domain **is all values except the ones that make the denominator become 0.

Ex. has a denominator that factors to (x –
4)(x + 3) and that makes the domain

become (−∞,−3)∪(−3, 4)∪(4,∞)which excludes –3 and 4.

2. For reasons on the video and in the text at the bottom of page 350, there are
“bumper lines” that

the graphs get very close to, but don’t cross over, and they are called**
asymptotes**.

A. The **Vertical Asymptote** is a line** x = a (number)**, where the graph approaches
infinity (goes out

the roof) on one side and approaches negative infinity (goes to the cellar) on
the other side, as the

values get closer and closer to the number. The values that were excluded in the
domain become the

Vertical Asymptotes. The Vertical Asymptote(s) are found by factoring the
denominator of the

function. Those values that we would exclude from the
domain since they would cause the

denominator to be = 0 become the values that are the Vertical Asymptote
equations . There may be

more than one, depending on the factoring.

Ex. The Vertical Asymptotes of the example in #1 are x = – 3 and x = 4. The
calculator is trying

to draw the Vertical Asymptotes in CONNECTED mode below.

Note that the “jags” in the graph happen at –3 and 4. |

B. The **Horizontal Asymptote** is the horizontal line y =** b (
number)** that the graph “bumps”.

Here are the three cases for finding a Horizontal Asymptote, if the rational
function has one. They

all depend on comparing the degrees of p(x) and q(x):

1. If the degree of the numerator polynomial is less than the degree of the
denominator polynomial,

then the x-axis or y = 0 is the horizontal axis. (Similar to a proper fraction
where the numerator is

less than the denominator)

Ex. The degree of each is:
, which is a “proper-degreed” situation, so
H.A. is y = 0.

2. If the degree of the numerator polynomial is the same as the degree of the
denominator

polynomial, then the line is the Horizontal
Asymptote.

Ex. . The degree of each is,
which is a tie. So the H. A. is .

3. If the degree of the numerator polynomial is greater than the degree of the
denominator

polynomial, then there is no horizontal asymptote. (Division by 0 in the video)

*We do not include anything that has to do with Oblique Asymptotes.

* The graph of a function never crosses a vertical asymptote.

* The graph of a function may sometimes cross a horizontal asymptote.

*Asymptotes are not part of the graph of the rational function, but are
necessary to give information

about the graph.

Graphing a Rational Function (p. 356 blue box): Here’s what you do-

1. Enter the function into the calculator, being careful to include ( ) around
any numerator or

denominator that needs it. This will give you a general idea of the turns that
the sketch will take.

2. Find the Vertical Asymptote(s) by finding the zeros of the denominator —factor
if possible, or use

quadratic formula (not on the test). Draw them on the graph as dotted lines
going up and down.

3. Find the Horizontal Asymptote, if there is one and sketch it as a dotted line
going left-right.

4. Find the y-intercept by evaluating the function at x = 0 (Substitute) or look
in TABLE.

5. Find the x-intercept(s) by finding the zeros of the numerator by solving p(x)
= 0 by factoring or

quadratic formula

6. Find other points from the TABLE to give it the general shape that you see in
the calculator

viewscreen.

If the function has a common factor between the numerator
and the denominator, then there will be a

“hole” at that value of the graph.

Ex. when reduced.

Don’t be fooled by what you see in the calculator on this
one! The domain of this function must still

exclude the –3 value that cannot happen. The domain is (−∞,−3)∪(−3,∞) . Please
make the “hole”

in your sketch a clearly open circle at the spot on the line in the first
quadrant where it would be.

*Note that reduced fraction doesn’t give a Vertical Asymptote and the “improper
degreed”

polynomials that make the function mean that there is no Horizontal Asymtote,
either.

II. Applications of Rational Functions

Ex. A drug is injected into a patient and the concentration of the drug in the
bloodstream is

monitored. The drug’s concentration, C(t), in milligrams per liter, after t
hours is modeled by

a. Sketch the graph using the graphing calculator. Do you need to see all of the
4 quadrants to work

this problem?

b. Name the Vertical Asymptote, the Horizontal Asymptote, the y-intercept and
the x-intercept.

Note that there may be an answer to these, or there may not be an answer.

c. What is the drug’s concentration after 3 hours?

d. Will the drug ever leave the patient’s bloodstream? (Hint: use what you know
about the

horizontal axis)

e. When did the patient receive the maximum amount of this drug?

(Answers on the next page.)

Assignments Text: pp. 362 – 367 #9 – 16, 29 – 34, 36, 37, 39, 43, 45, 81ab, 83abc |

Answers to the Application Example:

a.

Graph is [-3, 10, 1] x [-3, 5, 1] |

b. Vertical Asymptote: none, since x^{2} + 1 cannot be
factored.

Horizontal Asymptote: y = 0, since the fraction is a “proper degreed” one, where
the num < den

y-intercept: (0, 0)

x-intercept: (0, 0) only

c. The concentration after 3 hours can be found in the TABLE or by substitution .

The patient had approximately 1.5 milligrams per
liter of fluid in his or her bloodstream after 3 hours. |

d. The drug will never leave the patient’s bloodstream,
since there are no other x intercepts other

than the origin.

e. Use 2^{nd} CALC 4:Maximum.

The maximum dosage that the patient received was
after 1 hour, when he/she received 2.5 milligrams of medication. |

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