2-2 Polynomial Functions of Higher Degree

Polynomial The polynomial is written in standard form when the values of the exponents 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)=a n X n +a n-1 X n-1 +…+a 1 X+a 0

Polynomial in one variable Leading Coefficient

Polynomial POLYNOMIALEXPRESSIONDEGREELEADING COEFFICIENT Constant120 Linear4x-914 Quadratic5x 2 -6x-9 25 Cubic 8x 3 ＋ 12x 2 -3x ＋ 1 38 Generala n X n +a n-1 X n-1 +…+a 1 X+a 0 nanan

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

Kinds & Names by Degree 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 You have to find the term with the highest exponent if the polynomial is not in its standard form..

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,

Graphs of Polynomial Function

Synthetic Substitution

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Find if Function Values of Variables Replace x with 5e-2, 6 Simplify e

Practice

End behavior of a Polynomial function DEGREE LEADING COEFFICIENT EVENODD POSITIVE NEGATIVE

End Behavior I Even degree functionOdd degree function Degree: 4 Leading Coefficient: positive End behavior: f(x) →+ as x → - f(x) → + as x →+ Degree: 3 Leading Coefficient: positive End behavior: f(x) →- as x → - f(x) → + as x →+

End Behavior II Even degree functionOdd degree function Degree: 2 Leading Coefficient: negative End Behavior: f(x) → - as x→ - f(x) → - as x→ + Degree: 3 Leading Coefficient: negative End 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-degree 1 real zero Even-degree 2 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. DegreeLeading Coefficient Left end as x  - ∞ Right end as x  + ∞ OddPositive f(x)  - ∞ f(x)  + ∞ OddNegative f(x)  + ∞ f(x)  - ∞ EvenPositive f(x)  + ∞ EvenNegative 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 -> + 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→+ ∞, , f(x)→- ∞ as x→- ∞, f(x)→- ∞ as x→+ ∞, -4

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 ( 최댓값 ) or extreme maximum or extrema while the lowest point is the minimum( 최솟값 ), 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

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

ab 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 degree 3)Using the sign (+ or -) of the leading coefficient and the degree, you know the end behaviors 4)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