Continuity and End Behavior

of Polynomials

Continuity...

• All polynomial functions are continuous.

• Discontinuities – Three types:

Infinite (VA)

Point (Hole)

Jump

Continuity…• Polynomial Functions - Continuous

• Rational Functions – Discontinuity at…

2 5 4y x x 3 2y x 3 22 4 5 2y x x x

2 3 43 2 2y x x x x

2

3y

x

2

1

1

xy

x

Continuity…• Other continuous functions:

• Other functions with discontinuities:

1

1

xy

x

2

1

1y

x

y x

y x

Continuity… Piece-wise Functions

3 1

3 1

x for xy

x for x

3 0

2 4 0

x for xy

x for x

2 2

2 2

x for xy

x for x

End Behavior…

• Four Possibilities– Up on both ends

– Down on both ends

– Up on the right & Down on the left

– Up on the left & Down on the right

End Behavior…

Four Prototypes:– Up on both ends… y = x2

– Down on both ends… y = -x2

– Up on the right & Down on the left… y = x3

– Up on the left & Down on the right… y = -x3

Describe the End Behavior:

2 5 4y x x

3 2y x 3 22 4 5 2y x x x

2 3 43 2 2y x x x x

4 3 23 2 1y x x x

5 24y x x

End Behavior…

Notation:– Up on both ends…

– Down on both ends…

– Up on the right & Down on the left…

– Up on the left & Down on the right…

x y

x y

x y

x y

x y

x y

x y

x y