1. 3.1 Higher Degree Polynomial Functions and Graphs
an is called the leading coefficient
anxn is called the dominating term
a0 is called the constant term
P(0) = a0 is the y-intercept of the graph of P
Polynomial Function
A polynomial function of degree n in the variable x is a function defined by
where each ai is real, an  0, and n is a whole number.

2. 3.1 Cubic Functions: Odd Degree Polynomials
The cubic function is a third degree polynomial of the form
In general, the graph of a cubic function will resemble one of the following shapes.

3. 3.1 Quartic Functions: Even Degree Polynomials
The quartic function is a fourth degree polynomial of the form
In general, the graph of a quartic function will resemble one of the following shapes. The dashed portions indicate irregular, but smooth, behavior.

4. 3.1 Extrema
Turning points – where the graph of a function changes from increasing to decreasing or vice versa
Local maximum point – highest point or “peak” in an interval
function values at these points are called local maxima
Local minimum point – lowest point or “valley” in an interval
function values at these points are called local minima
Extrema – plural of extremum, includes all local maxima and local minima

5. 3.1 Extrema

6. 3.1 Number of Local Extrema
A linear function - degree 1 - no local extrema.
A quadratic function - degree 2 - one extreme point.
A cubic function - degree 3 - at most two local extrema.
A quartic function -degree 4 - at most three local extrema.
Extending this idea:
Number of Turning Points
The number of turning points of the graph of a polynomial function of degree n  1 is at most n – 1.

7. 3.1 End Behavior
Let axn be the dominating term of a polynomial function P.
n odd
If a positive, the graph of P falls on the left and rises on the right.
If a is negative, the graph of P rises on the left and falls on the right.
n even
If a > 0, the graph of P opens up.
If a < 0, the graph of P opens down.

8. 3.1 Determining End Behavior
Match each function with its graph.
Solution: B. A. C. D.
f matches C,
g matches A,
h matches B,
k matches D.

9. 3.1 Analyzing a Polynomial Function
Determine its domain.
Determine its range.
Use its graph to find approximations of local extrema.
Use its graph to find the approximate and/or exact
x- intercepts.
Solution
Since P is a polynomial, its domain is (–, ).
Because it is of odd degree, its range is (–, ).

10. 3.1 Analyzing a Polynomial Function
Two extreme points that we approximate using a graphing calculator: local maximum point (– 2.02,10.01), and
local minimum point (.41, – 4.24).
Looking Ahead to Calculus
The derivative gives the slope of f at any value in the domain. The
slope at local extrema is 0 since the tangent line is horizontal.

11. 3.1 Analyzing a Polynomial Function
(d) We use calculator methods to find that the x-intercepts are –1 (exact), 1.14(approximate), and –2.52 (approximate).

12. 3.1 Comprehensive Graphs
The most important features of the graph of a polynomial function are:
intercepts,
extrema,
end behavior.
A comprehensive graph of a polynomial function will exhibit the following features:
all x-intercepts (if any),
the y-intercept,
all extreme points (if any),
enough of the graph to exhibit end behavior.

13. 3.1 Determining the Appropriate Graphing Window
The window [–1.25,1.25] by [–400,50] is used in the
following graph. Is this a comprehensive graph?
Solution
Since P is a sixth degree
polynomial, it can have
up to 6 x-intercepts. Try
a window of [-8,8] by
[-1000,600].