Polynomials and End Behavior

Polynomial functions are classified by their degree. The graphs of polynomial functions are classified by

the degree of the polynomial. Each graph, based on degree has a distinctive shape and

characteristics.

• The degree of a polynomial is given by the term with the greatest degree. The leading coefficient is the coefficient of the first term when in standard form.

• The end behavior of a graph is a description of the values of the function as approaches positive infinity or negative infinity

x x

As x gets more and more negative, the graph of f(x) decreases, or approaches negative infinity.

As x gets more and more positive, the graph of f(x) increases, or approaches positive infinity.

x

x

A turning point is where a graph changes from increasing to decreasing or from decreasing to increasing (local

maximum or local minimum).

How many turning points does this graph have?

• The zeros of the graph are the values of x where the graph hits the x-axis.

Where are the zeros of the function?

3 24 4

( 1)( 1)( 4)

y x x x

y x x x

Graph of the equation: 2 1y x

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y1. What is the degree of this polynomial?

Odd or Even

2. How many turning points does this graph have?

3. Is the leading coefficient positive or negative?

4. As , f(x) approaches .

5. As , f(x) approaches .

6. How many zeros does the graph have?

x

x

Linear Function

-4

-1

0

1

4

y x

Quadratic Function 2y x

-2

-1

0

1

2

Cubic Function

-2

-1

0

1

2

3y x

Absolute-Value Function

-2

-1

0

1

2

y x

Square-Root Function

0

1

4

9

y x

Why didn’t I pick negative x-xalues?

Transformations

• Vertical translation

3

3(

)

3

(

)g x

f x x

x

3

3(

)

4

(

)g x

f x x

x

Transformations

• Horizontal Translation

3

3

2

)

) )

(

( (g x

f x

x

x

3

3

1

)

) )

(

( (g x

f x

x

x

Transformations

• Reflection across x-axis

3

3

( )

( )f

f x

x

x

x