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SFM Productions Presents:
Another day of Pre-Calculus torture!
No fun for you - tons of fon for me!
2.2 Polynomial Functions of Higher Degree
Homework for section 2.2
p145
9-16 all, 21-29, 37-49 eoo,
5563 eoo, 65-85 eoo, 97,99
A Polynomial function is continuous.
A Polynomial function has only smooth rounded turns.
If we take the simplest function, f(x) = xn
if n is even, then the graph touches the x-axis
(in certain cases, the graph doesn’t even have to touch the x-axis at all…)
if n is odd, then the graph goes through the x-axis
f(x)=x2 or
f(x) = x4 or
f(x) = x6
10
Y
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
X
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
f(x)=x3 or
f(x) = x5 or
f(x) = x7
10
Y
9
8
7
6
5
4
3
2
1
X
-3
-2.5
-2
-1.5
-1
-0.5
0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
0.5
1
1.5
2
2.5
3
What about more “challenging” functions:
f(x)= x4 + 3x3 - 2x + 5
and f(x)= -x4 - 3x3 + 2x + 5
10
Y
9.5
9
8.5
8
7.5
7
6.5
6
5.5
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
X
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
The Leading Coefficient test (and what it means)
If n is even
If n is odd
Y
Y
X
If a>0
X
Y
If a<0
X
Y
X
In the function f(x) , with some degree n :
The function has, at most, n-1 turning points.
The function has, at most, n real zeros
and speaking of REAL zeros………
REAL Zeros of Polynomial Functions
If f(x) is a polynomial function, and a is a real
number, the following things all mean the same thing.
a
x=a
x=a
is a root of the function f(x)
is a zero of the function f(x)
is a solution of the polynomial
equation f(x) = 0
(x-a) is a factor of the polynomial
f(x)
(a , 0)
is an x-intercept of the graph
of f
Sketching the graph of polynomial functions, or, why we’re doing all of this...
Sketch f(x) = -2x4 + 2x2
First, look at the leading coefficient and degree to determine what the graph is doing
at the extreme left and the extreme right.
Then, find the zeros of the function…
-2x4 + 2x2 = 0
-2x2(x2 - 1) = 0
X=0
X=±1
Note that x2 = 0 means
that x = 0 and x = 0.
∴ the repeated zero has a multiplicity of 2 (which is even)
because there are two of the same number
A multiplicity that is even means that the graph of the function touches the x-axis at that
zero.
A multiplicity that is odd means that the graph of the function goes through the x-axis at
that zero.
In the example from the previous slide, we have
a zero of even (2) multiplicity at 0 ,
and
a zero of odd (1) multiplicity at -1 ,
and
a zero of odd (1) multiplicity at 1.
And, since we remember (?) what the graph is doing at the extreme left and extreme right,
we can sketch a very reasonable graph of f(x) = -2x4 + 2x2
Y
X
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
f ( x )  2 x
S ket ch t he grap h of :
2 x
3
 6x
 6x
2
9

x
2
Rises left and
Falls right
Leading coefficient test tell us…
Zeros:
3
2
9

x  0
2
1

x 4 x 2  12 x  9   0
2
1

x 2 x  3
2
x  0,
Mult = 1

2
x 
(odd)
 0
3
,
2
x 
3
2
Mult = 2 (even)
5
Y
Inflection
Point
4.5
4
3.5
b
x  
3a
3
2.5
x  1
2
f ( x )  0 .5
1.5
1
0.5
X
-2
-1.5
-1
-0.5
0
-0.5
-1
-1.5
-2
0.5
1
1.5
2
Finding polynomials when given the zeros…
If x = 3 and x = 8, find a polynomial that fits.
If x = 3, then 3 is a zero (it is also a solution)
Same thing for x = 8.
That means that (x - 3) and (x - 8) are factors.
(x - 3)(x - 8) = 0
x2 - 11x + 24 = 0
2x2 - 22x + 48 = 0
3x2 - 33x + 72 = 0
a(x - 3)(x - 8) = 0
a could be any number, so there are an
infinite number of “correct” answers
All three have a different shape…due to different coefficients…which cause different stretches.
The Intermediate Value Theorem
10
9
8
7
6
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1-10
-2
-3
-4
-5
-6
-7
-8
-9
-10
Y
?
1 2 3 4 5 6 7 8 9 10
X
a  2
f ( a )  6 .8
b  4
f ( b)  6 .4
Somewhere, this graph has to cross zero…
Go! Do!