section 4.1 polynomial functions. a polynomial function is a function of the form a n, a n-1,…, a...
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Section 4.1Polynomial Functions
A polynomial function is a function of the form
f x a x a x a x an
n
n
n( )
1
1
1 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 x x( ) 3 4 52
Polynomial of degree 2
(b) h x x( ) 3 5
Not a polynomial
(c) F xx
x( )
3
5 2
5
Not a polynomial
A power function of degree n is a function of the form
nn xaxf )(
where a is a real number
a = 0
n > 0 is an integer.
2 1 0 1 2
2
4
6
8
10
y x 4
y x 8
(1, 1)(-1, 1)
(0, 0)
Power Functions with Even Degree
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.
2 1 0 1 2
10
6
2
2
6
10
y x 5
y x 9
(1, 1)
(-1, -1)
(0, 0)
Power Functions with Odd Degree
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.
4)1(2124)( 44 xxxf
y x 4
5 0 5
15
15
(0,0)
(1,1)
5 0 5
15
15
y x 2 4
(0,0)
(1, -2)
5 0 5
15
15
(1,0)
(2,-2)
y x 2 1 4
5 0 5
15
15
(1, 4)
(2, 2)
412 4 xy
If r is a Zero of Even Multiplicity
Graph crosses x-axis at r.
If r is a Zero of Odd Multiplicity
Graph touches x-axis at r.
For the polynomial f x x x x( ) 1 5 42
(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)
20)40)(50)(10()0( f
So, the y-intercept is (0,-20)
For the polynomial f x x x x( ) 1 5 42
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.
4)( xxf
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 f x x x x( ) 1 5 42
For the polynomial f x x x x( ) 1 5 42
On the interval 4 1x
Test number: x = -2
f (-2) = -14
Graph of f: Below x-axisPoint on graph: (-2, -14)
On the interval 1 5x
Test number: x = 0f (0) = -20
Graph of f: Below x-axis
Point on graph: (0, -20)
For the polynomial f x x x x( ) 1 5 42
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.
8 6 4 2 0 2 4 6 8
300
100
100
300
500(6, 490)
(5, 0)(0, -20)
(-1, 0)
(-2, -14)(-4, 0)
(-5, 160)
Sections 4.2 & 4.3Rational Functions
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A rational function is a function of the form
Rxpxqx
()()()
• 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.
(a) R xx
x x( )
1
8 122
x
x x1
6 2
All real numbers x except -6 and -2.
(b) R xx
x( )
4
162
x
x x4
4 4
All real numbers x except -4 and 4.
(c) R xx
( ) 5
92 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) 31
Example: Find the vertical asymptotes, if any, of the graph of each rational function.
(a) R xx
( ) 3
12
3
1 1( )( )x x
Vertical asymptotes: x = -1 and x = 1
(b) R xxx
( )
512
No vertical asymptotes
(c) R xx
x x( )
3
122
x
x x3
3 4( )( )
1
4x
Vertical asymptote: x = -432
(3,2)
(1,0)
(2,0)(0,1)
12
1)(
xxf
In this example there is a vertical asymptote at x = 2 and a horizontal asymptote at y = 1.
Examples of Horizontal Asymptotes
y = L
y = R(x)
y
x
y = L
y = R(x)
y
x
LxRx
)(lim
LxRx
)(lim
LxRx
)(lim
Examples of Vertical Asymptotes
x = cy
x
x = c
y
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.
Recall that the graph of isxxf
1)(
(1,1)
(-1,-1)
37
Graph the function using transformations
12
1)(
xxf
(1,1)
(-1,-1)
xxf
1)(
(3,1)
(1,-1)(2,0)
2
1)(
xxf
(3,2)
(1,0)
(2,0)(0,1)
011
1
011
1
)(
)()(
bxbxbxb
axaxaxa
xq
xpxR
mm
mm
nn
nn
Consider the rational function
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
(a) R xx x
x x x( )
3 4 15
4 7 1
2
3 2
Horizontal asymptote: y = 0
(b) R xx xx x
( )
2 4 13 5
2
2
Horizontal asymptote: y = 2/3
(c) R xx x
x( )
2 4 12
x x xx
x x
x
x
2 4 16
6 1
6 12
2
- 2
-
13
2
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.
42
Example: Analyze the graph of 9
642)(
2
2
x
xxxR
R x
x x
x x( )
2 2 3
3 3
2
2 3 13 3
x xx x
2 1
33
xx
x,
Domain: x x x 3 3,
a.) x-intercept when x + 1 = 0: (-1,0)
b.) y-intercept when x = 0: 3
2
)30(
)10(2)0(
R
y - intercept: (0, 2/3)
3
12)(
x
xxR
c.) Test for Symmetry: R xx
x( )
( )( )
2 13
)()()( xRxRxR No symmetry
R xx
xx( ) ,
2 1
33
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 x x3 3 1 1
x x x3 3 1 1
Test at x = -4
R(-4) = 6
Above x-axis
Point: (-4, 6)
Test at x = -2
R(-2) = -2
Below x-axis
Point: (-2, -2)
Test at x = 1
R(1) = 1
Above x-axis
Point: (1, 1)
g.) Finally, graph the rational function R(x)
8 6 4 2 0 2 4 6
10
5
5
10
(-4, 6)
(-2, -2) (-1, 0) (0, 2/3)
(1, 1) (3, 4/3)
y = 2
x = - 3