Graphs of polynomial functions

GRAPHS OF POLYNOMIAL FUNCTIONS

THE GRAPH OF POLYNOMIAL FUNCTIONS

The graph of a polynomial function has the following characteristics

SMOOTH CURVE - the turning points are not sharp CONTINUOUS CURVE – if you traced the graph

with a pen, you would never have to lift the pen The DOMAIN is the set of real numbers The X – INTERCEPT is the abscissa of the point

where the graph touches the x – axis. ABSOLUTE MAXIMUM/MINIMUM is the highest or

lowest point (respectively) of the graph of a polynomial function.

RELATIVE MAXIMUM/MINIMUM are the turning points of the graph of a polynomial function.

Let’s explore the graphs of the following functions

Let’s explore the graphs of the following functions

P(x) Degree (Odd/Eve

n)

Value of leading coefficie

nt

Rational zeros

Number of x-

intercepts

Number of

turning points

1 Odd a > 0 0 1 0

2 Odd a > 0 2, 4, 6 3 2

3 Odd a < 0 4 1 2

4 Even a > 0 0 1 1

5 Even a > 0 1, -1, 2, -2

4 3

6 Even a < 0 none 0 1

What conclusions can we draw from the table?

How would you relate number of turning points with the degree of each function?

What can be said about the number of zeros that each graph has and its relationship with the degree of its respective function?

What seems to be true with the graph’s behavior and its degree? the value of its leading coefficient?

Maximum Number of Turning Points and X-intercepts

A polynomial function of degree n hasa maximum number of n-1

turning pointsat most n x-intercepts

BEHAVIOR/TREND OF THE GRAPHS

Leading coefficient

Degree (Odd/Even

)Description of the

Graph

a > 0 Even Comes down from the left, goes up to the right

a > 0 Odd Comes up from the left, goes up to the right

a < 0 Even Comes up from the left, goes down to the right

a < 0 Odd Comes down from the left, goes down to the right

3)( xxf 3)( xxf

Let’s Practice

Describe the behavior of the following polynomial functions and identify the number of maximum zeros and turning points.

1.

2.

3.

3613)( 24 xxxf

xxxxxf 8822)( 234

3613)( 24 xxxf