2. Chapter 3: Polynomial Functions
3.1 Complex Numbers
3.2 Quadratic Functions and Graphs
3.3 Quadratic Equations and Inequalities
3.4 Further Applications of Quadratic Functions and Models
3.5 Higher-Degree Polynomial Functions and Graphs
3.6 Topics in the Theory of Polynomial Functions (I)
3.7 Topics in the Theory of Polynomial Functions (II)
3.8 Polynomial Equations and Inequalities; Further Applications and Models

3. 3.5 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.

4. 3.5 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.

5. 3.5 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.

6. 3.5 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

7. 3.5 Extrema

8. 3.5 Absolute and Local Extrema
Let c be in the domain of P. Then
(a) P(c) is an absolute maximum if P(c) > P(x) for all x in the domain of P.
(b) P(c) is an absolute minimum if P(c) < P(x) for all x in the domain of P.
(c) P(c) is a local maximum if P(c) > P(x) when x is near c.
(d) P(c) is a local minimum if P(c) < P(x) when x is near c.
The expression “near c” means that there is an open interval
in the domain of P conaining c, where P(c) satifies the
inequality.

9. 3.5 Identifying Local and Absolute Extrema
Example Consider the following graph.
Identify and classify the local extreme points of f.
Identify and classify the local extreme points of g.
Describe the absolute extreme points for f and g.
Local Min points: (a,b),(e,h); Local Max point: (c,d)
Local Min point: (j,k); Local Max point: (m,n)
f has an absolute minimum value of h, but no absolute maximum. g has no absolute extrema.

10. 3.5 Number of Local Extrema
A linear function has degree 1 and no local extrema.
A quadratic function has degree 2 with one extreme point.
A cubic function has degree 3 with at most two local extrema.
A quartic function has degree 4 with 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.

11. 3.5 End Behavior
Let axn be the dominating term of a polynomial function P.
n odd
If a > 0, the graph of P falls on the left and rises on the right.
If a < 0, 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.

12. 3.5 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.

13. 3.5 x-Intercepts (Real Zeros)
Number of x-Intercepts (Real Zeros) of a Polynomial Function
The graph of a polynomial function of degree n will have at most n x-intercepts (real zeros).
Example Find the x-intercepts of
Solution By using the
graphing calculator in a
standard viewing window,
the x-intercepts (real zeros)
are –2, approximately
–3.30, and approximately 0.30.

14. 3.5 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 (–, ).

15. 3.5 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 (0.41, – 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.

16. 3.5 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).

17. 3.5 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 reveal the correct end behavior.

18. 3.5 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].