Transcript Lesson 3.5A Rational Functions and their Graphs

Alg 2 Warm Up – Wed (5/15)-Thurs (5/16)
1.List the possible roots. Then find all the
zeros of the polynomial function.
f(x) = x4 – 2x2 – 16x -15
Answers:
Possibles: +/- 1, 3, 5, 15
{3, -1,-1+2i,-1-2i}
Announcements
Section 3.5 – Rational Functions and their Graphs
Objective:
Find the domain of rational functions.
Use arrow notation.
Find the vertical and horizontal asymptotes.
Graph Rational Functions
3.5 – Finding the domain of a Rational
Function
To find the domain set the denominator to zero
and solve for x. The domain will be all real
number except that value for x. x 2  9
The denominator is x- 3.
x 3
Solve for x.
x- 3 = 0
x=3
The domain is all Real Numbers
Except x≠3
Example 2
Find the domain:
x3
x2  9
X2 + 9 = 0
X2 = -9
The domain is only for real numbers not
imaginary the domain is(-∞, ∞)
YOU TRY!!!
Find the domain of
x
x 2  25
x3
Find the domain of 2
x  16
Answers
1. All real numbers except
x≠5 or -5
2. All real numbers.
Reciprocal Function
The reciprocal function is the most basic
rational function. It is f(x) = 1/x.
It looks like:
If x is far from zero, then 1/x is
close to zero.
If x is close to zero, then 1/x is far
from zero.
Arrow Notation
How to use Arrow Notation
If you have x
3- means what is the graph
doing as x approaches 3 from the left. Left
because of the negative attached.
Example
Let’s look at problem 9
It is asking to look as x
approaches -3 from the left
what is the graph doing?
Look at the graph from the
left and the first curve is
going to -∞ so the answer
is f(x)
-∞.
You Try #13
Answer: f(x) is
approaching
zero.
The parent function of rational functions is
1
f ( x) 
x
What does the graph look like?
Another Basic Rational Function
f(x) = 1/x2 and it looks like this:
Asymptotes: lines that a graph approaches
but does not cross
Vertical asymptotes:
 Whichever values are not allowed in the
domain will be vertical asymptotes on the
graph.
Where is the domain limited?
denominator
Set those factors that only appear in the
denominator or those that appear more
times in the denominator than numerator
equal to zero and solve.
Definition of Vertical Asymptote
Example
3x  1
1.Find the vertical asymptotes: f ( x) 
x2
Set x – 2 = 0, x = 2 is a vertical asymptote.
x3
2.Find the vertical asymptotes:f ( x )  2
x 9
2
Factor x – 9 = (x-3)(x+3)
x3
1

f ( x) 
x 3
( x  3)( x  3)
Set x – 3 = 0, x =3 is the vertical asymptote
There won’t be one at x=-3, which means there is a
hole in the graph at -3 or point discontinuity.
Picture next slide.
Picture
If the factors are not real, then we say no
vertical asymptotes.
Horizontal asymptotes:
 Look at the degrees of the numerator and
denominator
If the degrees are equal then the horizontal
asymptote is the ratio of the leading coefficients
( y = ratio of leading coefficients)
If the degree in the denominator is greater then
the horizontal asymptote is y = 0
If the degree in the numerator is greater then
there is no horizontal asymptote.
Definition of Horizontal Asymptotes
Ex 1: Find the horizontal asymptotes of
each rational function
 A)
3x  1
f ( x) 
x2
4x
B) f ( x)  2 x 2  1
Horizontal: degrees
Horizontal: If the degree
in the denominator is
greater than the
numerator horizontal
asymptote is y = 0
1st
are equal (both are
degree) so y = ratio 3/1
y=3
C)
4x3
f ( x)  2
2x  1
If the degree in the numerator is
greater than there is no horizontal
asymptote.
Pictures of A,B, and C
YOU TRY!!! Ex 1: find the vertical &
horizontal asymptotes of each rational
function
 A)
4x  1
f ( x) 
x5
 Vertical: x-5=0
x=5
Horizontal: degrees
are equal (both are 1st
degree) so y = ratio 4/1
y=4
2
x
 3x  10
 B) f ( x) 
( x  1)( x  2)( x  3)
 Vertical:
x+1 = 0
x+3=0
So x = -1, x = -3,
Horizontal: denominator is bigger
(3rd degree vs. 2nd)
so y
=0
x = -2 is a hole
Lesson 3.5 Graphing a Rational
Function
Rational Functions that are not
transformations of f(x) = 1/x or f(x) =
1/x2 can be graphed using the following
suggestions.
Strategy for Graphing a Rational Function
The following strategy can be used to graph f(x) = p(x)
q(x),
Where p and q are polynomial functions with no common
factors.
Seven steps:
Determine whether the graph of
f has symmetry.
f(-x) = f(x): y-axis symmetry
f(-x) = -f(x): origin symmetry
 Step 1:
•Step 2: Find the y-intercept (if there is one) by evaluating
f(0).
•Step 3: Find the x-intercepts (if there are any) by solving
the equation p(x) = 0.
Steps continued:
 Step 4: Find any vertical asymptote(s) by solving
the equation q(x) =0.
•Step 5: Find the horizontal asymptote (if there is one )
using the rule for determining the horizontal asymptote
of a rational function.
•Step 6: Plot at least one point between and beyond
each x-intercept and vertical asymptote.
•Step 7: Use the information obtained previously to
graph the function between and beyond the vertical
asymptotes.
Example 1: Graph:
f(x) = 2x
x-1
Example 2:
2
3x
f ( x)  2
x 4
Step 1: Determine Symmetry: f(-x)
Step 2: Find the Y-intercept. f(0)
Step 3: Find the x-intercepts. p(x) = 0
Step 4: Find the vertical asymptote(s). q(x)=0
Step 5: Find the horizontal asymptote. (Degree of numerator and denominator)
Step 6: Plot points between and beyond each x-intercept and vertical asymptote.
(table)
Step 7: Graph the function.
2
3x
f ( x)  2
x 4
3( x)
3x
1. f ( x) 
 2
2
(  x)  4 x  4
2
2. f(0) = 3*02 =
02 – 4
2
0 = 0
The graph of f is symmetric with
respect to the y-axis.
The y-intercept is 0, so the graph
passes thru the origin.
-4
3. 3x2=0 , so x = 0.
The x-intercepts is 0, verifying
the graph passes through the
origin.
4. Set q(x) =0,
x2-4
= 0, so x = 2 and
x= -2
The vertical asymptotes are x =
-2 and x=2.
Ex. 2 cont:
5. Look at the degree of numerator
and denominator. They are equal
so you use the leading
coefficients. 3/1
The horizontal
asymptote is y = 3.
6. Plot points between and beyond each x-intercept and
vertical asymptote.
X
-3 -1
f(x) 3x2 27/ -1
x2 -4 5
1 3
4
-1 27/5 4
7. Graph the functions.
Summary:
If you are given the equation of a rational
function, explain how to find the vertical
asymptotes of the function.