Transcript Sullivan College Algebra Section 4.1

Section 4.1
Polynomial Functions
A polynomial function is a function of the
form
f ( x )  a n x n  a n  1 x n  1   a 1 x  a 0
an , an-1 ,…, a1 , a0 are real numbers
n is a nonnegative integer
D: {x|x å real numbers}
Degree is the largest power of x
Example: Determine which of the following are
polynomials. For those that are, state the degree.
(a) f ( x )  3 x 2  4 x  5
Polynomial of degree 2
(b) h ( x )  3 x  5
Not a polynomial
5
3x
(c) F ( x ) 
5  2x
Not a polynomial
A power function of degree n is a function
of the form
f ( x)  a n x
where a is a real number
a =0
n > 0 is an integer.
n
Power Functions with Even Degree
10
yx
8
8
yx
4
6
4
2
(-1, 1)
2
(1, 1)
1
(0, 0)
0
1
2
Summary of Power Functions with Even Degree
1.) Symmetric with respect to the y-axis.
2.) D: {x|x is a real number}
R: {x|x is a non negative real number}
3.) Graph (0, 0); (1, 1); and (-1, 1).
4.) As the exponent increases, the graph increases
very rapidly as x increases, but for x near the
origin the graph tends to flatten out and lie closer
to the x-axis.
Power Functions with Odd Degree
10
yx
9
6
2
(-1, -1)
1
yx
5
2
(0, 0)
2
0
6
10
(1, 1)
1
2
Summary of Power Functions with Odd Degree
1.) Symmetric with respect to the origin.
2.) D: {x|x is a real number}
R: {x|x is a real number}
3.) Graph contains (0, 0); (1, 1); and (-1, -1).
4.) As the exponent increases, the graph becomes
more vertical when x > 1 or x < -1, but for -1 < x
< 1, the graphs tends to flatten out and lie closer to
the x-axis.
Graph the following function using transformations.
f ( x )  4  2 x  1  2( x  1)  4
4
4
15
15
(1,1)
5
(0,0) 0
5
15
yx
(0,0)
5
0
(1, -2)
15
4
y  2 x 4
5
15
15
(1, 4)
(1,0)
5
0
(2, 2)
5
(2,-2)
5
0
5
15
15
y  2 x  1
4
y  2x 1  4
4
If r is a Zero of Even Multiplicity
Graph touches
x-axis at r.
If r is a Zero of Odd Multiplicity
Graph crosses
x-axis at r.
For the polynomial
2
f ( x )   x  1  x  5 x  4
(a) Find the x- and y-intercepts of the graph of f.
The x intercepts (zeros) are (-1, 0), (5,0), and (-4,0)
To find the y - intercept, evaluate f(0)
f (0)  (0  1)(0  5)(0  4)  20
So, the y-intercept is (0,-20)
For the polynomial
2
f ( x )   x  1  x  5 x  4
b.) Determine whether the graph crosses or touches
the x-axis at each x-intercept.
x = -4 is a zero of multiplicity 1 (crosses the x-axis)
x = -1 is a zero of multiplicity 2 (touches the x-axis)
x = 5 is a zero of multiplicity 1 (crosses the x-axis)
c.) Find the power function that the graph of f
resembles for large values of x.
f (x)  x
4
For the polynomial
2
f ( x )   x  1  x  5 x  4
d.) Determine the maximum number of turning points
on the graph of f.
At most 3 turning points.
e.) Use the x-intercepts and test numbers to find the
intervals on which the graph of f is above the x-axis and
the intervals on which the graph is below the x-axis.
On the interval    x   4
Test number:
x = -5
f (-5) = 160
Graph of f: Above x-axis
Point on graph: (-5, 160)
For the polynomial
2
f ( x )   x  1  x  5 x  4
On the interval  4  x   1
Test number:
x = -2
f (-2) = -14
Graph of f: Below x-axis
Point on graph: (-2, -14)
On the interval  1  x  5
Test number:
f (0) = -20
Graph of f:
x= 0
Below x-axis
Point on graph: (0, -20)
For the polynomial
2
f ( x )   x  1  x  5 x  4
On the interval 5  x  
Test number:
x=6
f (6) = 490
Graph of f: Above x-axis
Point on graph: (6, 490)
f.) Put all the information together, and connect the
points with a smooth, continuous curve to obtain the
graph of f.
500
(6, 490)
300
(-1, 0)
(-5, 160)
100
8
(-4, 0)
6
4
(0, -20)
2 0 2
100
(-2, -14)
300
4
(5, 0)
6
8
Sections 4.2 & 4.3
Rational Functions
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A rational function is a function of the form
p( x)
R( x ) 
q( x)
• p and q are polynomial functions
• q is not the zero polynomial.
• D: {x|x å real numbers & q(x) = 0}.
Find the domain of the following rational functions.
x 1
x

1
(a) R ( x )  2

x  8 x  12
 x  6 x  2
All real numbers x except -6 and -2.
x4
x4

(b) R ( x )  2
 x  4 x  4
x  16
All real numbers x except -4 and 4.
5
(c) R ( x )  2
x 9
All Real Numbers
30
Vertical Asymptotes.
Domain gives vertical asymptotes
•Reduce rational function to lowest terms, to find
vertical asymptote(s).
•The graph of a function will never intersect vertical
asymptotes.
•Describes the behavior of the graph as x approaches
some number c
Range gives horizontal asymptotes
•The graph of a function may cross intersect
horizontal asymptote(s).
•Describes the behavior of the graph as x
approaches infinity or negative infinity (end
behavior)
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Example: Find the vertical asymptotes, if any, of the
graph of each rational function.
3
3

(a) R ( x )  2
x  1 ( x  1)( x  1)
Vertical asymptotes: x = -1 and x = 1
x 5
(b) R ( x )  2
x 1
No vertical asymptotes
1
x3
x3


(c) R ( x )  2
x4
x  x  12 ( x  3)( x  4)
Vertical asymptote: x = -4
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(3,2)
(0,1)
(2,0)
(1,0)
1
f (x) 
1
x2
In this example there is a
vertical asymptote at x = 2
and a horizontal asymptote
at y = 1.
lim R( x)  L
x 
Examples of Horizontal Asymptotes
y
y = R(x)
y=L
x
lim R( x)  L
x 
y
y=L
x
y = R(x)
Examples of Vertical Asymptotes
x=c
y
x=c
y
x
x
If an asymptote is neither horizontal nor vertical
it is called oblique.
y
x
Note: a graph may intersect it’s oblique
asymptote. Describes end behavior. More on this
in Section 3.4.
1
Recall that the graph of f ( x)  x
is
(1,1)
(-1,-1)
37
1
Graph the function f ( x )  x  2  1 using transformations
(1,1)
(3,1)
(2,0)
(1,-1)
(-1,-1)
f ( x) 
1
x
1
f ( x) 
x2
(3,2)
(0,1)
(2,0)
(1,0)
Consider the rational function
p ( x ) an x n  an 1 x n 1    a1 x  a0
R( x) 

q ( x ) bm x m  bm 1 x m 1    b1 x  b0
1. If n < m, then y = 0 is a horizontal asymptote
2. If n = m, then y = an / bm is a horizontal asymptote
3. If n = m + 1, then y = ax + b is an oblique asymptote,
found using long division.
4. If n > m + 1, neither a horizontal nor oblique
asymptote exists.
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Example: Find the horizontal or oblique
asymptotes, if any, of the graph of
3x  4 x  15
(a) R ( x )  3
2
x  4x  7x  1
2
Horizontal asymptote: y = 0
2 x2  4 x  1
(b) R ( x ) 
3x 2  x  5
Horizontal asymptote: y = 2/3
x  4x  1
(c) R ( x ) 
x2
2
x6
x  2 x2  4 x  1
-  x 2  2 x
6x  1
- 6 x  12
13
Oblique asymptote: y = x + 6
To analyze the graph of a rational function:
1) Find the Domain.
2) Locate the intercepts, if any.
3) Test for Symmetry. If R(-x) = R(x), there is
symmetry with respect to the y-axis. If - R(x) =
R(-x), there is symmetry with respect to the origin.
4) Find the vertical asymptotes.
5) Locate the horizontal or oblique asymptotes.
6) Determine where the graph is above the x-axis
and where the graph is below the x-axis.
7) Use all found information to graph the function.
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2x2  4x  6
Example: Analyze the graph of R ( x ) 
x2  9
2 x  2 x  3
R( x) 
 x  3 x  3
2
2 x  3 x  1

 x  3 x  3
2 x  1

, x3
 x  3
Domain:
 x x  3, x  3
2 x  1
R( x) 
 x  3
a.) x-intercept when x + 1 = 0:
(-1,0)
2 ( 0  1) 2

b.) y-intercept when x = 0: R ( 0 ) 
( 0  3)
3
y - intercept: (0, 2/3)
c.) Test for Symmetry:
2(  x  1)
R(  x) 
(  x  3)
R(x)  R(x)  R(x)
No symmetry
2 x  1
R( x) 
, x3
 x  3
d.) Vertical asymptote: x = -3
Since the function isn’t defined at x = 3, there is
a hole at that point.
e.) Horizontal asymptote: y = 2
f.) Divide the domain using the zeros and the
vertical asymptotes. The intervals to test are:
   x  3
 3  x  1
1 x  
   x  3  3  x  1  1  x  
Test at x = -4
Test at x = -2
Test at x = 1
R(-4) = 6
R(-2) = -2
R(1) = 1
Above x-axis
Below x-axis
Above x-axis
Point: (-4, 6)
Point: (-2, -2)
Point: (1, 1)
g.) Finally, graph the rational function R(x)
x=-3
10
(-4, 6)
5
(1, 1)
(3, 4/3)
y=2
8
6
(-2, -2)
4
2
0
5
10
2
(-1, 0)
4
(0, 2/3)
6