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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