Transcript Document

Sullivan Algebra and
Trigonometry: Section 5.1
Polynomial Functions
Objectives
• Identify Polynomials and Their Degree
• Graph Polynomial Functions Using Transformations
• Identify the Zeros of a Polynomial and Their Multiplicity
• Analyze the Graph of a Polynomial Function
A polynomial function is a function of the
form
f ( x )  a n x  a n 1 x
n
n 1
  a 1 x  a 0
where an , an-1 ,…, a1 , a0 are real numbers
and n is a nonnegative integer. The domain
consists of all real numbers. The degree of
the polynomial is the largest power of x that
appears.
Example: Determine which of the following are
polynomials. For those that are, state the degree.
(a) f ( x )  3x  4 x  5
2
Polynomial of degree 2
(b) h ( x )  3 x  5
Not a polynomial
5
3x
(c) F ( x ) 
5  2x
Not a polynomial
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
f ( x )  ( x  1)( x  4)
Consider the polynomial:
2
Solve the equation f (x) = 0
2
f ( x )  ( x  1)( x  4) = 0
x+1=0
OR
x-4=0
x=-1
OR
x=4
If f is a polynomial function and r is a real number
for which f (r) = 0, then r is called a (real) zero of
f, or root of f. If r is a (real) zero of f, then
a.) (r,0) is an x-intercept of the graph of f.
b.) (x - r) is a factor of f.
x  r 
m
m 1


x

r
is a factor of a polynomial f and
If
is not a factor of f, then r is called a zero of
multiplicity m of f.
Example: Find all real zeros of the following
function and their multiplicity.
5
1
2


f ( x )   x  3  x  7 x  

2
x = 3 is a zero with multiplicity 2.
x = - 7 is a zero with multiplicity 1.
x = 1/2 is a zero with multiplicity 5.
If r is a Zero of Even Multiplicity
Sign of f (x) does not
change from one side to
the other side of r.
Graph touches
x-axis at r.
If r is a Zero of Odd Multiplicity
Sign of f (x) changes
from one side to the
other side of r.
Graph crosses
x-axis at r.
Theorem: If f is a polynomial function of
degree n, then f has at most n - 1 turning
points.
Theorem: For large values of x, either
positive or negative, the graph of the
polynomial
f ( x )  a n x  a n 1 x
n
n 1
  a 1 x  a 0
resembles the graph of the power function
f ( x)  a n x
n
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
(0, -20)
4 100
2 0 2
(-2, -14)
300
4
(5, 0)
6
8