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Graphs of polynomial functions
1.
2. The graph of a polynomial function has the following
characteristics
ïš SMOOTH CURVE - the turning points are not sharp
ïš CONTINUOUS CURVE â if you traced the graph with a
pen, you would never have to lift the pen
ïš The DOMAIN is the set of real numbers
ïš The X â INTERCEPT is the abscissa of the point where
the graph touches the x â axis.
ïš ABSOLUTE MAXIMUM/MINIMUM is the highest or
lowest point (respectively) of the graph of a polynomial
function.
ïš RELATIVE MAXIMUM/MINIMUM are the turning points
of the graph of a polynomial function.
3.
4.
5. Value of Number of
P(x) Degree leading Rational Number of turning
(Odd/Even coefficient zeros x- points
) intercepts
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
6. ïš How would you relate number of turning
points with the degree of each function?
ïš What can be said about the number of zeros
that each graph has and its relationship
with the degree of its respective function?
ïš What seems to be true with the graphâs
behavior and its degree? the value of its
leading coefficient?
7. ïš A polynomial function of degree n has
ïĄ a maximum number of n-1 turning
points
ïĄ at most n x-intercepts
8. Leading Degree
coefficient (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
15. ïš Describe the behavior of the following
polynomial functions and identify the number
of maximum zeros and turning points.
4 2
1. f ( x) x 13x 36
4 3 2
2. f ( x) 2x 2x 8x 8x
4 2
3. f ( x) x 13x 36