3.5 Higher – Degree Polynomial Functions and Graphs
Polynomial Function
Definition: A polynomial function of degree n in the variable x is a function defined by
Where each ai(0 ≤ i ≤ n-1) is a real number, an ≠ 0, and n is a whole number. What’s the domain of a polynomial function?
P(x) = anxn + an-1xn-1 + … + a1x + a0
Get to know a polynomial function
P(x) = anxn + an-1xn-1 + … + a1x + a0
an : Leading coefficient
anxn : Dominating
term
a0 : Constant term
Cubic Functions
P(x) = ax3 + bx2 + cx + d
(b)(a)
(d)(c)
Quartic Functions
P(x) = ax4 + bx3 + cx2 + dx + e
(b)(a)
(d)(c)
ExtremaTurning points: points where the
function changes from increasing to decreasing or vice versa.
Local maximum point: the highest point at a peak. The corresponding function values are called local maxima.
Local minimum point: the lowest point at a valley. The corresponding function values are called local minima.
Extrema: either local maxima or local minima.
Absolute and Local ExtremaLet 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 an local maximum if P(c) ≥ P(x) when x is near c.
(d) P(c) is an local minimum if P(c) ≤ P(x) when x is near c.
Example
Local minimum point
Local minimumpoint
Local minimum &Absolute minimumpoint
Local minimum point
Local minimum point
A function can only have one and only one
absolute minimum of maximum
Hidden behavior
Hidden behavior of a polynomial function is the function behaviors which are not apparent in a particular window of the calculator.
Number of Turning PointsThe number of turning points of
the graph of a polynomial function of degree n ≥ 1 is at most n – 1.
Example: f(x) = x f(x) = x2
f(x) = x3
End BehaviorDefinition: The end behavior of a
polynomial function is the increasing of decreasing property of the function when its independent variable reaches to ∞ or - ∞
The end behavior of the graph of a polynomial function is determined by the sign of the leading coefficient and the parity of the degree.
End Behavior
Odd degree
a > 0
a < 0
Even degree
a > 0
a < 0
exampleDetermining end behavior Given
the Polynomial f(x) = x4 –x2 +5x -4
X – Intercepts (Real Zeros)Theorem: The graph of a
polynomial function of degree n will have at most n x-intercepts (real zeros).
Example: P(x) = x3 + 5x2 +5x -2
Comprehensive GraphsA comprehensive graph of a
polynomial function will exhibit the following features:
1. all x-intercept (if any) 2. the y-intercept 3. all extreme points(if any)
4. enough of the graph to reveal the correct end behavior
example1. f(x) = 2x3 – x2 -22. f(x) = -2x3 - 14x2 + 2x + 84 a) what is the degree? b) Describe the end behavior of the
graph. c) What is the y-intercept? d) Find any local/absolute maximum
value(s). ... local/absolute maximum points. [repeat for minimums]
e) Approximate any values of x for which f(x) = 0
HomeworkPG. 210: 10-50(M5), 60, 63
KEY: 25, 60
Reading: 3.6 Polynomial Fncs (I)
