3.5 higher – degree polynomial functions and graphs

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)