polynomial behavior patterns in the graphs. warm up list and name the transformations in this...

Polynomial Behavior

patterns in the graphs

Warm Up

• List and name the transformations in this diagram

Standard Form of Polynomial

Degree Name using

degree

Polynomial example

Number of terms

Name using # terms

0 5 1

1 X + 4 2

2 4x2 1

3 4x3 - 3x2 + x 3

4 2x4 + 5x2 2

5 -x5 +4x2 +2x +1 4

Lesson 4.4 Graphing Polynomial Functions

Key Terms:• Relative Max / Min• Real Zeros (x-intercepts)

Example 1 Graphing Polynomials

A) xxxxxf 44)( 234 B) 54)( 23 xxxf

Zeros: 1, -1.4, -3.6

R. Min: (-2.7, -4.5)

R. Max: (0, 5)

Left & Right End Behavior

Degree 0 Degree 2

Degree 3 Degree 4

Degree 1

Degree 2

Degree 3 Degree 4

Degree 1Negative

Polynomials

Positive Polynomials

Studying Graphs of Polynomials

OddDegree 1Positive LC

EvenDegree 2Positive LC

OddDegree 3Positive LC

EvenDegree 4Positive LC

OddDegree 5Positive LC

OddDegree 5Negative LC

EvenDegree 4Negative LC

Key Notes:[1] Odd when rt & lf ends go opposite directions

[2] Even when rt & lf ends go in same direction

[3] Positive right end goes up

[4] Negative right end goes down

[5] Degree is one more than the number of turns

Example 3 Graphs of Polynomial Functions

Determine if the polynomial is even/odd, Positive / Negative, find the Degree, and state the number of Real Zeros / Imaginary.

a)

Even

Negative

Degree 2

2 Real Zeros

b)

Odd

Positive

Degree 5

1 Real / 4 Imaginary Zeros

Determine if the polynomial is even/odd, positive/negative, and give the degree.

c) d)

Even

Negative

Degree 4

Odd

Positive

Degree 5

Power Functions are any function of the form  f(x) = axn

• where a and n are nonzero constant real numbers

• n is the exponent;

1.If If n = positive integer (linear, quadratic, cubic, etc)

2.If n = negative integer (rational function)

3.If n = fraction (square root function)

Power Functions are any function of the form  f(x) = axn

• If If n = positive integer (linear, quadratic, cubic, etc)

f(x) = ax3 f(x) = ax4

Power Functions are any function of the form  f(x) = axn

• If n = negative integer (rational function)

f(x) = ax-1 = 1/x

Power Functions are any function of the form  f(x) = axn

1. If n = fraction (square root function)

f(x) = ax1/2

Homework 4.4

1. a) b) Zeros:

c) Rel. Max: Rel. Min:

– 4 , 0

(0, 0)(-2.7, -9.5)

Warm-Up

yxxyx 33 5312

xxxf 2)(

209

4168

)4()4()4(

2

2

2

xx

xxx

xxxf

[1] State the degree and LC:

[2] , find )4( xf

Degree 4LC = 5

Example 2 Polynomial Applications

A) Suppose you have a 12 x 14 sheet of cardboard. You plan to cut a uniform corner from each corner and fold the sheet intoan open box. - What is the maximum volume?- What are the dimensions of the maximized box?

B) Use a 10 x 12 sheet of cardboard

10 – 2x

12 – 2xx

)212)(210( xxxV

Max V: 96. 8 units3

Dimensions: 1.8 x 6.4 x 8.4

160. 6 units3

2.1 x 7.7 x 9.7