2 important ideas..
1.) End behavior
2.) Where the function
crosses the x-axis(zeros)
Think ...
the first term
is important!
first term
3
1
x=1, x= -1, x = 3
set each factor = 0
first term
x= 0, x= -4 , x= 5, x= 3, x= 3
5
.08
The Graph "bounces" off x = 3.
first term
4
2 Factors
x =3 (times 2)
Multiplicity!
x= 2, x = -5, x = -5, x = -5
-0.05
The Graph is "flattening" at x = -5
3 Factors
Multiplicity!
x =-5 (times 3)
5
x= -6(times 3), x = 1(times 2)
-0.02
the graph
"bounces"
the graph
"flattens out"
3
2
x=0, x =-3, x = 4(times 2)
4
-0.05
x=0, 1 factor...passes through in a linear
fashion
x =-3, 1 factor...passes through in a linear
fashion
x = 4
2 factors, bounces at the root
x=1(times 3),x= -3, x= 5(times 2)
6
0.002
x=1, 3 factors...flattens at the root
x =-3, 1 factor...passes through in a linear
fashion
x = 5
2 factors, bounces at the root
When a factor is squared,
the graph "bounces" at the root.
When a factor is cubed,
the graph "flattens" at the root.
First term
4
1
x= 0, x =3, x = -2, x= -2
Bounce at x = -2 . Passing through x =0, and x =3
in a linear fashion.
First term
4
-1
x=0(times 2),x =2, x= -3
x =0 ...bounce
x =2 ...passing through...
x= -3...passing through...
When a factor is raised to the first power,
the graph "passes through" through the root
in a linear fashion.
First term
6
1
x = 0, x =3(times 3), x = -2( times 2)
x=0 ....passing through in a linear fashion
x=3 ....flattens out at the root
x=-2....bounces at the root
4
4
x =0, x= 0, x =-2, x=2
The graph "bounces" at x =0, and passes through both x = -2 and x =2
in a linear fashion.
Key
2 important ideas..
1.) End behavior
2.) Where the function
crosses the x-axis(zeros)
Think ...
the first term
is important!
first term
3
1
set each factor = 0
x=1, x= -1, x = 3
first term
5
x= 0, x= -4 , x= 5, x= 3, x= 3
.08
The Graph "bounces" off x = 3.
first term
4
2 Factors
Multiplicity!
x =3 (times 2)
x= 2, x = -5, x = -5, x = -5
-0.05
The Graph is "flattening" at x = -5
3 Factors
Multiplicity!
x =-5 (times 3)
Key
5
x= -6(times 3), x = 1(times 2)
-0.02
the graph
"bounces"
the graph
"flattens out"
3
2
x=0, x =-3, x = 4(times 2)
4
-0.05
x=0, 1 factor...passes through in a linear
fashion
x =-3, 1 factor...passes through in a linear
fashion
x = 4
2 factors, bounces at the root
x=1(times 3),x= -3, x= 5(times 2)
6
0.002
x=1, 3 factors...flattens at the root
x =-3, 1 factor...passes through in a linear
fashion
x = 5
2 factors, bounces at the root
Key
When a factor is squared,
the graph "bounces" at the root.
When a factor is cubed,
the graph "flattens" at the root.
First term
4
1
x= 0, x =3, x = -2, x= -2
Bounce at x = -2 . Passing through x =0, and x =3
in a linear fashion.
First term
4
-1
x=0(times 2),x =2, x= -3
x =0 ...bounce
x =2 ...passing through...
x= -3...passing through...
When a factor is raised to the first power,
the graph "passes through" through the root
in a linear fashion.
First term
6
1
x = 0, x =3(times 3), x = -2( times 2)
x=0 ....passing through in a linear fashion
x=3 ....flattens out at the root
x=-2....bounces at the root
4
4
x =0, x= 0, x =-2, x=2
The graph "bounces" at x =0, and passes through both x = -2 and x =2
in a linear fashion.
Key
Hw p. 293; 13 - 18 all