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)
b)
c)
d)
e)
f)
g)
h)
i)
j)
k)
l)
m)
n)
y = x2 + 2x + 1
y =-2x2 + 3x – 1
y = -x2
y = 2x2 – 1
y = 2x3 + 4x – 1
y = -x3 + 2x – 1
y = 4x4 + 1
y = 3x3 – 2
y = 5x5
y = -4x5 + 1
y = 8x8
y = -4x7 + 5x3 – 4x – 1
y = 8x3 – 4x4 – 2x + 3
y = -3x5 + 4x6 – x4
End Behavior
Given f ( x)  ax n  bx n1  ...  ex  f
Right Hand Side
Left Hand Side
lim f ( x)
x 
lim f ( x)
x 
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)
Multiplicity
When 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!
Sketch
1) y = x3 + x2 – x – 1
Sketch
2) y = x4 – 8x2 + 16
Sketch
3) y = 2x5 + x4 – 6x3
Sketch
4) y = -2(x + 2)2(x – 2)
Sketch
5) y = 3(x – 1)3(x + 2)2(x – 4)