Post on 14-Dec-2015
Zeros and End Behavior
Objective: Be able to find zeros and end behavior of a graph.
TS: Making decisions after reflection and review
Exploration:
a) Find the end behavior of each of the following equations. Look for a pattern and then describe how you would find the end behavior without actually graphing.
b) Find the number of x-intercepts of each of the equation. See if you can find a connection between the number of x-intercepts and the equation itself.
a) y = x2 + 2x + 1b) y =-2x2 + 3x – 1 c) y = -x2
d) y = 2x2 – 1 e) y = 2x3 + 4x – 1f) y = -x3 + 2x – 1 g) y = 4x4 + 1h) y = 3x3 – 2 i) y = 5x5
j) y = -4x5 + 1k) y = 8x8
l) y = -4x7 + 5x3 – 4x – 1 m) y = 8x3 – 4x4 – 2x + 3n) y = -3x5 + 4x6 – x4
End Behavior
Right Hand Side
Left Hand Side
lim ( )x
f x
lim ( )x
f x
1( ) ...n nGiven f x ax bx ex f
Find the zeros of each of the following.
1) y = x3 + x2 – x – 1
2) y = x4 – 8x2 + 16
3) y = 2x5 + x4 – 6x3
4) y = -2(x + 2)2(x – 2)
5) y = 3(x – 1)3(x + 2)2(x – 4)
MultiplicityWhen a zero is repeated it is said to have a multiplicity equal to degree
of it’s factor.
Which zeros from the last examples have a multiplicity, and what are their multiplicites?
What happens to the graph when a zero has a multiplicity?
..
Graph sketching
When you’re asked to sketch a graph…
1) Find the end behavior
2) Find the zeros and their multiplicities
3) Find the y-intercept
4) Sketch away!