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