Transcript chapter 4 notes, 

Introduction to Rational
Functions and their graphs
Definition and Domain
College Algebra chapter 4
Introduction to Rational Functions and their graphs
A rational function is a function which is the ratio of polynomial
functions. Said differently, r is a rational function if it is of the form
p( x)
r ( x) 
, where p and q are polynomial functions
q( x)
Note : All polinomials are rational functions for q ( x )  1.
Remember that we have domain issues anytime the denominator
of a fraction is zero. In the next examples, we review this concept
as well as some of the arithmetic of rational expressions.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Find the domain of the following rational functions.
p( x)
Write them in the form
q( x)
for polynomial functions p and q and simplify.
Setting x  1  0 results in x  1.
Hence, our domain is ( , 1)  ( 1, ).
The expression f ( x) is already in the form
requested and when we check for common
factors among the numerator and denominator
we find none, so we are done.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
p( x)
Find the domain and write them in the form
, and simplify.
q( x)
The denominators in the formula for h( x) are both x 2 -1 whose zeros are x  1.
As a result, the domain of h is ( , 1)  ( 1,1)  (1, ).
College Algebra chapter 4
Introduction to Rational Functions and their graphs
p( x)
Find the domain and write them in the form
, and simplify.
2
q( x)
2x 1
2
x
1 .
It may help to temporarily rewrite r ( x) as r ( x) 
3x  2
x2  1
We need to set all of the denominators equal to zero
which means we need to solve not only
x 2  1  0  x  1 ,but also
2
3x  2
 0  3x  2  0  x  .
2
3
x 1
2
2
Our domain is : (, 1)  (1, )  ( ,1)  (1, )
3
3
College Algebra chapter 4
Introduction to Rational Functions and their graphs
We simplify r ( x) :
Lesson: Do Not Simplify Before Finding The Domain! In this case we would have lost 1 and
-1 if we had simplified before we found the domain!
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Vertical Asymptotes
College Algebra chapter 4
Introduction to Rational Functions and their graphs
The line x  c is called a vertical asymptote of
the graph of a function y  f ( x)


if as x  c , or as x  c ,
then either f ( x)   or f ( x)  .
In other words :
p( x)
r ( x) 
has vertical symptotes at x-values
q( x)
that make q( x) (but not p ( x)) zero.
College Algebra chapter 4
Example
Introduction to Rational Functions and their graphs
x  2x 1
r ( x) 
x 1
x  1  0  x  1 is
the vertical asymptote.
Note : x  1 does not make
the numerator zero.
3
College Algebra chapter 4
Introduction to Rational Functions and their graphs
X-intercepts
College Algebra chapter 4
Introduction to Rational Functions and their graphs
p( x)
r ( x) 
has x-intercepts
q( x)
at x-values that make p( x) (but not q( x)) zero.
For example :
( x  1)( x  2)
r ( x) 
3
x 1
has x  int( s ) : x  1 and x  2
because 1 and  2 make the numerator zero.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Example
( x  1)( x  2)( x  3)
r ( x) 
( x  2)
( x  1)( x  2)( x  3)  0
 x  1, x  2, x  3
are x-intercepts.
Note : None of the x-values
x  1, x  2, x  3
makes the denominator zero.
x  2  0  x  2 is the vertical asymptote.
Note: x  2 does not make the numerator zero.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Holes in the graph
College Algebra chapter 4
Introduction to Rational Functions and their graphs
p( x)
r ( x) 
has holes at x-values
q( x)
that make p( x) AND q( x) zero.
( x  2)( x  3)( x  4)
Example : r ( x) 
x( x 2  x  6)
x  2, x  3, and x  4 make the numerator zero,
but only x  4 is the x  int
r ( x) 
( x  2) ( x  3) ( x  4)
x ( x  3) ( x  2)
because x  2, x  3 also make the denominator zero.
In fact the graph has holes at 2 and -3.
x  0 is the vertical asymptote.
College Algebra chapter 4
( x  4)

x
Introduction to Rational Functions and their graphs
Example
( x  1)( x  2)( x  3)
r ( x) 
( x  2)( x  1)
( x  1)( x  2)( x  3)  0
 x  1, x  2, x  3,
but only x  2, x  3 are x-intercepts.
Note : x  1 makes the denominator zero.
So, there is a hole at x  1.
( x  2)( x  1)  0  x  2 , x  1,
but only x  2 is a vertical asymptote.
Note: x  2 does not make the numerator zero.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Horizontal and
Polynomial Asymptotes
College Algebra chapter 4
Introduction to Rational Functions and their graphs
The line y  c is called a horizontal asymptote of the graph
of a function y  f ( x) if as x   or as x  , f ( x)  c.
p ( x) an x 
r ( x) 

m
q ( x) bm x 
n
 a0
has :
 b0
an
---A horizontal asymptote y 
if n  m.
bm
---A polynomial asymptote y  L( x) where L( x)
is the quotient obtained by dividing p( x) by q ( x) if n  m.
---A horizontal asymptote y  0 if n  m.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Example
( x  1)( x  2)( x  3) ( x  2)( x  3)
r ( x) 

( x  2)( x  1)
( x  2)
( x  1)( x  2)( x  3)  0  x  1, x  2, x  3,
but only x  2, x  3 are x-intercepts.
Note : x  1 makes the denominator zero.
So, there is a hole at x  1.
Note : r (1)  6 and hole is (1,6).
College Algebra chapter 4
Example
( x  1)( x  2)( x  3)
r ( x) 
( x  2)( x  1)
Introduction to Rational Functions and their graphs
( x  2)( x  1)  0  x  2 , x  1,
but only x  2 is a vertical asymptote.
Note: x  2 does not make the numerator zero.
x  x6
4
Canceling ( x  1) we get
 ( x  1) 
.
x2
( x  2)
So, y  x  1 is the polynomial (slant, if n  m  1) asymptote.
2
College Algebra chapter 4
Example
( x  1)( x  2)( x  3)
r ( x) 
( x  2)( x  1)
Introduction to Rational Functions and their graphs
Note: We know that the graph
goes to infinity at x=2. To find
out whether it goes to – or +
infinity we pick a number close
to 2 from the left of 2 and one
from the right of 2, then we
check the sign of r(x) by
substituting for x.
Choose 2.1 and sub it for x in r(x).The sign of
r(2.1) is (+)(+)(-)/(+)(+) negative. So, r(x) goes
to negative infinity on the left of 2. Do the
same for x=1.9. r(1.9) is positive, so r(x) goes
to positive infinity on the left of 2.
College Algebra chapter 4
Example
( x  1)( x  2)( x  3)
r ( x) 
( x  2)( x  1)
Introduction to Rational Functions and their graphs
Does it intersect the slant
asymptote? We check this by
setting r(x)=x+1, and see if
there are solutions:
( x  2)( x  3)
 ( x  1)  x 2  x  6  x 2  x  2
( x  2)
6  2  No solutions !
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Example
x  2x  x
r ( x)  2
x x2
3
2
x  2x  x
x( x  1)
x( x  1)


2
x x2
( x  2)( x  1) ( x  2)
3
2
College Algebra chapter 4
2
Introduction to Rational Functions and their graphs
Example
x  2x  x
r ( x)  2
x x2
3
2
x  2x  x
x( x  1)
x( x  1)


2
x x2
( x  2)( x  1) ( x  2)
3
2
Does it intersect the slant
asymptote? We check this by
setting r(x)=x+3, and see if
there are solutions:
x( x  1)
2
2
 ( x  3)  x  x  x  x  6
( x  2)
0  6  No solutions !
College Algebra chapter 4
2
Introduction to Rational Functions and their graphs
Example: Suppose the cost C to make x players is C ( x)  100 x  2000, x  0.
---Interpret the behavior of C ( x) as x  0.
As x → 0+, average cost → ∞.
This means that as fewer and
fewer players are produced,
the cost per player becomes
unbounded. In this situation,
there is a fixed cost of $2000
(C(0) = 2000), we are trying to
spread that $2000 over fewer
and fewer players.
College Algebra chapter 4
100 x  2000
C ( x) 
x
Introduction to Rational Functions and their graphs
Example: Suppose the cost C to make x players is C ( x)  100 x  2000, x  0.
---Interpret the behavior of C ( x) as x  .
As x→∞, Avg.cost → 100+. This means that as more and more
players are produced, the cost per player approaches $100, but is
always a little more than $100.
C ( x) 
College Algebra chapter 4
100 x  2000
x
Introduction to Rational Functions and their graphs
Example: The number of students N at local college who have had
the flu t months after the semester begins can be modeled by the
given formula below for t ≥ 0.
450
N (t )  500 
1  3t
1.
N (0)  500 
450
 50
1  3(0)
Find and interpret N(0).
450
450
 300 
 200
1  3t
1  3t
5
450  200(1  3t )  t  .
12
500 
2. How long will it take until 300 students
will have had the flu?
This means that at the beginning
of the semester, 50 students
have had the flu.
3. Determine the behavior of N as t → ∞.
Interpret this result graphically and within
the context of the problem.
This means it will take 5 /12
months, or about 13 days, for 300
students to have had the flu.
t → ∞, N(t) → 500. This
means as time goes by, only
a total of 500 students will
have ever had the flu.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Example: The number of students N at local college who have had
the flu t months after the semester begins can be modeled by the
given formula below for t ≥ 0.
450
N (t )  500 
1  3t
1. Find and interpret N(0).
450
N (0)  500 
 50
1  3(0)
This means that at the
beginning of the semester, 50
students have had the flu.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Example: The number of students N at local college who have had
the flu t months after the semester begins can be modeled by the
given formula below for t ≥ 0.
450 2. How long will it take until 300 students
N (t )  500 
1  3t will have had the flu?
This
means
it
will
450
450
500 
 300 
 200 take 5 /12
1  3t
1  3t
months, or about
5
450  200(1  3t )  t  .
13 days, for 300
12
students to have
had the flu.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Example: The number of students N at local college who have had
the flu t months after the semester begins can be modeled by the
given formula below for t ≥ 0.
450
N (t )  500 
1  3t
3. Determine the
behavior of N as t → ∞.
Interpret this result
graphically and within
the context of the
problem.
t → ∞, N(t) → 500. This means as
time goes by, only a total of 500
students will have ever had the flu.
College Algebra chapter 4
Introduction to Rational Functions and their graphs
Steps for Graphing Rational Functions
Suppose r is a rational function.
1. Find the domain of r.
2. Reduce r(x) to lowest terms, if applicable.
3. Find the x- and y-intercepts of the graph of y = r(x), if they exist.
4. Determine the location of any vertical asymptotes or holes in the
graph, if they exist. Analyze the behavior of r on either side of the
vertical asymptotes, if applicable.
5. Analyze the end behavior of r. Find the horizontal or polynomial
asymptote.
6. Use a sign diagram and plot additional points, as needed, to
sketch the graph of y = r(x).
College Algebra chapter 4
x2  2 x  1
Example : Graph f ( x)  3
2
x  x  2x
Introduction to Rational Functions and their graphs
x  2x 1
( x  1)( x  1)
( x  1)


3
2
x  x  2 x x( x  2)( x  1) x( x  2)
2
College Algebra chapter 4
x2  2 x  1
Example : Graph f ( x)  3
2
x  x  2x
Introduction to Rational Functions and their graphs
Note : the graph crosses its own H . A.:
x 1
 0  x 1  0  x  1
x( x  2)
College Algebra chapter 4
Variation
College Algebra chapter 4
Variation
Suppose x, y and z are variable quantities. We say:
1) y varies directly with (or is directly proportional to) x if
there is a constant k such that y = kx.
2) y varies inversely with (or is inversely proportional to) x if
there is a constant k such that y = k/x .
3) z varies jointly with (or is jointly proportional to) x and y if
there is a constant k such that z = kxy.
The constant k in the above definitions is called the constant
of proportionality.
College Algebra chapter 4
Variation
Translate the following into mathematical equations
using the definition above:
Hooke’s Law: The force F exerted on a spring is directly
proportional the extension x of the spring.
Applying the definition of direct variation, we get
F = kx for some constant k.
College Algebra chapter 4
Variation
Translate the following into mathematical equations
using the definition above:
Boyle’s Law: At a constant temperature, the pressure P of an
ideal gas is inversely proportional to its volume V .
Since P and V are inversely proportional, we write:
k
P
V
College Algebra chapter 4
Variation
Ohm’s Law: The current I through a conductor between two
points is directly proportional to the voltage V between the
two points and inversely proportional to the resistance R
between the two points.
Even though the problem doesn’t use the phrase
‘varies jointly’, it is implied by the fact that the current I
is related to two different quantities. Since I varies
directly with V but inversely with R, we write:
kV
I
R
College Algebra chapter 4
Variation
Translate the following into mathematical equations using the
definition above:
Newton’s Law of Universal Gravitation:
Suppose two objects, one of mass m and one of mass M, are
positioned so that the distance between their centers of mass is r.
The gravitational force F exerted on the two objects varies directly
with the product of the two masses and inversely with the square of
the distance between their centers of mass.
GmM
F
2
r
G  6.7  10
11
2
Nm
kg 2
College Algebra chapter 4
Rational Equations and Inequalities
College Algebra chapter 4
Rational Equations and Inequalities
Solve the rational equation. Be sure to check for extraneous
solutions.
The only point of inter
2
1
1
x 3
section
of
the
two
rational

 2
functions is at x=-1.
x3 x3 x 9
2
1
1
x
3 2
2
2
( x  9) 
( x  9)  2
( x  9)
x3
x3
x 9
( x  3)  ( x  3)  x 2  3
2x  x  3  x  2x  3  0
( x  3)( x  1)  0  x  3, x  1
But , x  3 is not in the domain.
The solution set  {1}.
2
2
College Algebra chapter 4
To solve a rational inequality:
Rational Equations and Inequalities
1)Turn it into an equation and solve.
2) Graph the solutions on a number line (use open
circles if the solution does not make the inequality
true, and filled in if it does). Also, include numbers
that make the denominator zero using open circles.
3) Test a number from each region on the number line
to see if the region is included in the solution set
(that is if the test number makes the inequality
true).
College Algebra chapter 4
x  5x  6
Example :
0
2
x 1
2
Rational Equations and Inequalities
x  5x  6
2
 0, x  1  x  5 x  6  0
2
x 1
 ( x  3)( x  2)  0  x  3, x  2
2
The solution set: (−∞,−3) ∪ (−2,−1) ∪ (1,∞)
College Algebra chapter 4
Rational Equations and Inequalities
Carl and Mike start a 3 mile race at the same time. If Mike ran the race at 6 miles
per hour and finishes the race 10 minutes before Carl, how fast does Carl run?
Time
Rate
Distance
Carl
t
x
3
Mike
t-(10/60)
6 miles/h
3
3
 6t  1  3  6t  4  t  4  2
tx  3  x 

6 3
t

3
3
3 9
miles
1
x   x   3    4.5

2
6(t  )  3  6t  1  3
t
2 2
hour

6
3
College Algebra chapter 4
Rational Equations and Inequalities
Working together, Daniel and Donnie can clean the llama pen in 45 minutes. On
his own, Daniel can clean the pen in an hour. How long does it take Donnie to
clean the llama pen on his own?
Time (minutes)
Rate
How much of the
work done
Daniel
60
1/60
1
Donnie
t
1/t
1
Together
45
1/45
1
1 1 1
60
4
 
 t  60  t  t  60  t
60 t 45
45
3
4
1
t  t  60  t  60  t  180 minutes  3 hours
3
3
College Algebra chapter 4