November 30, 2011
Warm Ups Alg 2 12-3
1. Evaluate
2. Simplify
a. (-2x5y3z-4)-2
c. (4x3)2
b.
d. (4 + x3)2
November 30, 2011
#12-2 p. 333?
November 30, 2011
Yesterday we saw what a polynomial was, and how to evaluate it.
There were two methods: direct substitution and synthetic
substitution. But ONLY if it is a polynomial and we use the latter
method.
f(x) = 5x3 + x2 + 1,
f(4) = ?
Show the two ways.
November 30, 2011
Today we will look at the graphs of polynomial equations. We
already know how to graph lines and quadratics.
DEGREE
0
1
2
TYPE
constant
linear
STANDARD FORM
f(x) = a
f(x) = ax + b
quadratic f(x) = ax2+bx+c
EXAMPLE
f(x) = 3
f(x) = 2x + 3
f(x) = 2x2-7x-4
November 30, 2011
We are familiar with those graphs. Describe their"end behavior"
Positive leading coefficient
degree 0
as x →-∞
f(x) →
as x →∞
f(x) →
Negative leading coefficient
degree 0
as x →-∞
f(x) →
as x →∞
f(x) →
November 30, 2011
Positive leading coefficient
degree 1
as x →-∞ as x →∞
f(x) →
f(x) →
Negative leading coefficient
degree 1
as x →-∞ as x →∞
f(x) →
f(x) →
November 30, 2011
Positive leading coefficient
Negative leading coefficient
degree 2
degree 2
as x →-∞ as x →∞
f(x) →
f(x) →
as x →-∞ as x →∞
f(x) →
f(x) →
November 30, 2011
But what about higher order functions?
DEGREE TYPE
3
4
5
6
STANDARD FORM
EXAMPLE
Describe end behavior of these graphs. I'll graph them on my
grapher-you record results.
November 30, 2011
Positive leading coefficient
Negative leading coefficient
degree 3
degree 3
as x →-∞ as x →∞
f(x) →
f(x) →
as x →-∞ as x →∞
f(x) →
f(x) →
November 30, 2011
Positive leading coefficient
Negative leading coefficient
degree 5
degree 5
as x →-∞ as x →∞
f(x) →
f(x) →
as x →-∞ as x →∞
f(x) →
f(x) →
November 30, 2011
Positive leading coefficient
Negative leading coefficient
degree 4
degree 4
as x →-∞ as x →∞
f(x) →
f(x) →
as x →-∞ as x →∞
f(x) →
f(x) →
November 30, 2011
Positive leading coefficient
Negative leading coefficient
degree 6
degree 6
as x →-∞ as x →∞
f(x) →
f(x) →
as x →-∞ as x →∞
f(x) →
f(x) →
November 30, 2011
SUMMARY To describe the end behaviour of
n
f(x) = anx + .......
for an > 0 and n is even,
as x →-∞
f(x) →
as x →∞
f(x) →
for an < 0 and n is even,
as x →-∞
f(x) →
as x →∞
f(x) →
for an > 0 and n is odd,
as x →-∞
f(x) →
as x →∞
f(x) →
for a < 0 and n is odd
n
as x →-∞
f(x) →
as x →∞
f(x) →
example:
November 30, 2011
Sketch the graphs
f(x) = -x3 + 2
f(x) = 3 - x4
f(x) = x3 + 2x2 - 2
November 30, 2011
#12-3 p 334 49 - 52, 54 - 81 /3
November 30, 2011