2-2 Polynomial Functions of Higher Degree

Polynomial

• The polynomial is written in standard form when the values of the expo-nents are in “descending order”.

• The degree of the polynomial is the value of the greatest exponent.

• The coefficient of the first term of a polynomial in standard form is called the leading coefficient.

• f(x)=anXn+an-1Xn-1+…+a1X+a0

Polynomial in one variable

LeadingCoefficient

PolynomialPOLYNOMIAL EXPRESSION DEGREE LEADING

COEFFICIENT

Constant 12 0 12

Linear 4x-9 1 4

Quadratic 5x2-6x-9  2 5

Cubic 8x3＋ 12x2-3x ＋ 1 3 8

General anXn+an-1Xn-1+…+a1X+a0

n an

A bouncy ball’s height can be modeled by a function . Find the height of the ball when the time is 2 sec.

Evaluate a Polynomial Function

Replace t with 2.

Calculate.

Practice

Calculate if

Calculate if

Calculate if

Kinds & Names by De-gree

• Constant function - 0• Linear function - 1• Quadratic function - 2• Cubic function - 3• Quartic function - 4• Quintic function - 5

The # of their solutions match with their degree!!

The # of their solutions match with the # of the degree or number of directions the graph travels or

at most (n-1) number of turns

Leading Coefficients

• The coefficient of the term with the highest exponent : 8 : 13 : 6

658 24 xx

1524713 368 xxx

You have to find the term with the high-est exponent if the polynomial is not in its standard form..

64568 275 xxx

Polynomial – monomial or sum of monomialsPolynomial function

≠0 , n≥0 is called the leading coefficient is the constant termStandard form – all exponents are in descending order.

Polynomial Function

• A continuous function that can be described by a polynomial equation in one variable

• Power Function: function form of when a and b are real numbers

• Evaluate it!– When ,

baxxf )(

532)( 2 xxxf195)2(3)4(2)2( f

Graphs of Polynomial Function

<constant> <linear> <quadratic>

<cubic>

<quartic>

Synthetic Substitution

• You can evaluate f(x) for a certain value, x, by using synthetic substitution.

f(x) = + +1To find f(4) you could substitute 4 for x and do lots of math or you can use synthetic substitution.

Synthetic Substitution

f(x) = + +1f(4)= 4 | 2 -2 -1 3 0(x term) 1 |______8___24__92___380 1520_

2 6 23 95 380 1521

f(4) = 1521

(4×2 )(4×6 )(4×23 )(4×95 )

Find f(2),

57

Find if

Function Values of Vari-ables

Replace x with 5e-2, 6Simplify

e

Practice

End behavior of a Polynomial function

DE-GREE

LEADINGCOEFFICIENT

EVEN ODD

POSITIVE

NEGA-TIVE

End Behavior I

Even degree function Odd degree function

Degree: 4Leading Coefficient: positiveEnd behavior: f(x) →+ as x → -

f(x) → + as x →+

Degree: 3Leading Coefficient: positiveEnd behavior: f(x) →- as x → -

f(x) → + as x →+

End Behavior II

Even degree function Odd degree function

Degree: 2Leading Coefficient: negativeEnd Behavior: f(x) → - as x→ -

f(x) → - as x→ +

Degree: 3Leading Coefficient: negativeEnd Behavior: f(x) → + as x → -

f(x) → - s x → +

Real Zeros

• Every even-degree function has an even number of real zeros and odd-degree function has odd number of real zeros.

Odd-degree1 real zero

Even-degree2 real zeros

Even-Degree Polynomials• The ends(where the x

approaches positive infinity or negative infinity) point in the same direction

• Have even number of real solutions

(or none)

quadratic function

quartic function

Odd-Degree Polynomials

linear function

cubic function quintic function

• Ends point at different directions

• Have odd number of real solutions

Why can even-degree functions have no real solutions but odd-degree polynomials can not:

The ends of odd-degree functions point at different directions so at least one end is bound

to cross the x-axis, which is the solution

End Behavior of Polynomials

• The end behavior of a graph is how the graph behaves at the ends.

Degree Leading Coefficient

Left endas x-∞ Right end

as x+∞Odd Positive f(x)-∞ f(x)+∞Odd Negative f(x)+∞ f(x)-∞Even Positive f(x)+∞ f(x)+∞Even Negative f(x)-∞ f(x)-∞

End Behavior

: Behavior of the graph of f(x) as x approaches positive infinity or negative infinity

F(x) ->- as x -> -F(x) -> + as x -> +

F(x) -> + as x -> -F(x) -> - as x -> +

F(x) -> + as x -> -F(x) -> + as x ->+

F(x) -> - as x -> -F(x) -> - as x -> +

The ends of even degree polynomial function graphs point in the same directions so f(x) always ends negative or always positive even

Practice Questions

• Tell the degree, end behavior and leading coefficient for each function.

1. 7, f(x)→- ∞ as x→-∞, f(x)→+∞ as x→+∞, 6

2. 6, f(x)→- ∞ as x→- ∞, f(x)→- ∞ as x→+ ∞, -4

3665)( 273 xxxxf

56584)( 236 xxxxxf

Graphs of Polynomial ( 다항식 ) Functions

Relative Maximums and Minimums

Where the graph changes direction is a local or relative maximum or minimum since no other points near it are larger for maximums or smaller for minimums.

The highest point is called the maximum ( 최 댓 값 ) or extreme maximum or extrema while the lowest point is the minimum(최 솟 값 ), extreme minimum or extrema.

All minima and maxima (plurals of minimum and maximum) are also called turning points since the graph “turns”.

The graph of a polynomial function of degree n has at most n-1 turning points.

Maximum and Minimum Points

Point A: Relative Maximum (no other nearby points have a greater y-coordinate)

Point B: Relative Minimum (no other nearby points have a lesser y-coordinate)

A

B

Relative or local Extrema

Turning Points

Location Principle

Suppose y = f(x) represents a polynomial function and a and b are two real numbers such that f(a)<0 and f(b)>0. Then the function has at least one real zero between a and b.

ba

f(a)

f(b)

There is a real zero between two points on the graph where the y values have opposite signs.

If is a function is continuous on a closed interval [a,b], and d is a number between f(a) and f(b), then there exists c ∈ [a,b] such that f(c) = d.If a function is continuous from [1,10] then and a number d (y-value) which is between f(1) and f(10) then there exists a number between 1 and 10, c, where f(c )=d

Intermediate Value Theorem

Intermediate Value Theorem

a b

f(a)

f(b)

d

c

How to sketch the Graph of a Polynomial Function

1) Look at the leading coefficient (positive or negative)

2) Determine the degree3) Using the sign (+ or -) of the leading coefficient

and the degree, you know the end behaviors4) Find any zeros if you can factor the polynomial

(Zero Product Property) 5) Use the rational zeros test to try and determine

other zeros (by dividing section 2.3) and/or plot points by making a table of values.

graph

1) y=

2) f(x) =