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!

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)