Degrees of Polynomials; End Behavior Unit 2 (2.2 Polynomial Functions)

Degrees of Polynomials; End

BehaviorUnit 2 (2.2 Polynomial Functions)

Warm-Up Go over homework from last night

Review on Quadratic Functions

Find the vertex, zeros andthen graph:

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Objectives

Students will be able to determine the end behavior of the graph of a polynomial function

Students will be able to, algebraically and using a calculator, find the zeros of a polynomial

Students will be able to graph a polynomial function based on its end behavior and zeros


Continuous Graphs Only smooth rounded curves Leading Coefficient Test Zeros Max and Min Increasing and Decreasing

Basic Characteristics of Polynomial Functions

y

x

–2

2


Graphing With someone at your table, graph different

polynomial equations of higher degree and graph them on your calculator. Do a degree for 3, 4, 5, 6, 7 and 8. Example: Graph , etc.

What patterns do you see?


What does the degree do?

All polynomials of even degree look something like

All polynomials of odd degree look something like

The higher exponents add “bumps” to the graph.

2)( xxf

3)( xxf


How a function acts as x gets really big or really small.

What does the function approach as x approaches infinity?

Also known as right-hand and left-hand behavior!

End Behavior


Leading Coefficient Test Used to determine the end behavior of the graph of

a polynomial function Leading coefficient – the number in front of the highest

exponent

Degree – the highest exponent

Examples:Find the leading coefficient and degree of each polynomial function.

Polynomial Function Leading Coefficient Degree

5 3( ) 2 3 5 1f x x x x 3 2( ) 6 7f x x x x

( ) 14f x

– 2 5

1 3

14 0


Leading Coefficient Test

4 cases

Even Exponent

Odd Exponent

Positive

Negative


End Behavior Describe the end behavior of these functions.

1.

2.

3.

xx 24

1 3

154.335

xx

2632

xx


Real Zeros of Polynomial Functions A real number a is a zero of a function y = f (x) if

and only if f (a) = 0. If y = f (x) is a polynomial function and a is a real

number then the following statements are equivalent. x = a is a zero of f. x = a is a solution of the polynomial equation f (x) = 0. (x – a) is a factor of the polynomial f (x). (a, 0) is an x-intercept of the graph of y = f (x).


Multiplicity

When a root is repeated (or used) in the polynomial

If the multiplicity is even, the graph will “bounce” off the x-axis at the root and return towards the direction it came from

If the multiplicity is odd, the graph will go through the x-axis at the root


Find the zero’s of the following functions

Solve for the zeros algebraically and then check using your calculator!


Example:

Find all the real zeros of

The real zeros arex = , x = , and x =

These correspond to the ________________

y

x

–2

2

f (x) = x4 – x3 – 2x2

(–1, 0) (0, 0)

(2, 0)


Zero’s with Calculator

Input function in y= Graph the function 2nd Calc Zero Left bound and enter Right bound and enter Guess and enter

TI-84


Closure

Graph the polynomial:


Homework

Textbookpage 148, #1-8 and 13-21 odd


Degrees of Polynomials; End Behavior Unit 2 (2.2 Polynomial Functions)

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