polynomial functions - andrews.edu

POLYNOMIAL FUNCTIONSAlgebra 2, Chapter 4

Text: Big Ideas Algebra 2

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Transum Mathematics

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PREPARE FOR CH. 4Objective:

I can add and subtract polynomials

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Adding and subtracting polynomials

Combine like terms

3(2x + 3) + 4(x – 2) (4x2 + x - 5) - (2x3 + 3x2 - 4x + 2)

Pg. 151 #1; pg. 152 #1-10

3(2x + 3) + 4(x – 2)6x + 9 + 4x – 8 10x – 1

(4x2 + x – 5) – (2x3 + 3x2 – 4x + 2)4x2 + x – 5 – 2x3 – 3x2 + 4x – 2- 2x3 + x2 + 5x – 7

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Transum Mathematics

Ans: 2n + 6Possible pairs: 2(n+3); -2(-n-3); 6(1/3n+1); etcMatthew 18 Parable of the lost sheep

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4.1 GRAPHING POLYNOMIAL FUNCTIONSObjectives:

I can identify and evaluate polynomial functions.

I can graph polynomial functions.

I can describe end behavior of polynomial functions.

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4.1 GRAPHING POLYNOMIAL FUNCTIONS

Work with a partner

Use a calculator and graph each function

Describe the end behavior (What is happening

at the left and right ends of the graph?)

Identify the term with the greatest exponent.

How does the exponent affect the graph?

𝑓 𝑥 = 𝑥 + 1

g 𝑥 = 3𝑥2 + 4𝑥 + 1

h 𝑥 = −𝑥3 − 1

𝑖 𝑥 = −𝑥4 + 3𝑥2 + 2

F(x): falls, risesG(x): rises, risesH(x): rises, fallsI(x): falls, falls

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Polynomial Function

Of the form f(x) = an xn + an-1 xn-1 + ⋅ ⋅ ⋅ + a1 x + a0

an ≠ 0; exponents are whole numbers (positive); coefficients are real.

Polynomial in one variable

Function that has one variable and there are powers of that

variable and all the powers are positive.

Degree

Highest power of the variable.

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Is the function a polynomial in one variable? If so, what is the degree?

2𝑥3 + 5𝑥 + 8

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𝑥

−47𝑥5932 − 32𝑥4278 + 1

4𝑥12 + 3𝑥𝑦 + 2

2𝑥3 + 5𝑥 + 8 yes; 3

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𝑥no

−47𝑥5932 − 32𝑥4278 + 1 yes, 5932

4𝑥12 + 3𝑥𝑦 + 2 no

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Common Types of Polynomial Functions

Degree Type Example

0 Constant 𝑓 𝑥 = 1

1 Linear 𝑓 𝑥 = 3𝑥 + 2

2 Quadratic 𝑓 𝑥 = 2𝑥2 − 𝑥 + 3

3 Cubic 𝑓 𝑥 = 𝑥3 + 𝑥2 + 4𝑥 − 1

4 Quartic 𝑓 𝑥 = −5𝑥4 + 2𝑥 + 3

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Direct Substitution

Plug in for all x’s

Evaluate 𝑓 𝑥 = 2𝑥3 + 3𝑥2 − 5𝑥 + 8 when x = 2

Replace all x’s with 2F(2) = 2(2)^3 + 3(2)^2 - 5(2) + 8= 2(8) + 3(4) – 10 + 8= 16 + 12 – 10 +8= 26

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End Behavior

The behavior of the graph as x approaches positive infinity or negative

infinity.

Determined by the degree and sign of the leading coefficient.

Write: 𝑓 𝑥 → −∞ 𝑎𝑠 𝑥 → −∞ 𝑎𝑛𝑑𝑓 𝑥 → −∞ 𝑎𝑠 𝑥 → +∞

Leading Coefficient + Leading Coefficient -

Even Degree

Odd Degree

Even, negative

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Graphing

Make a table of values

Plot the points

Draw the curve

Check end behaviors

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Graph 𝑓 𝑥 = 𝑥3 + 𝑥2 − 4𝑥 + 2

Pg. 158 # 1, 5, 7, 9, 13, 15, 19, 21, 23, 25, 29,

31, 33, 35, 39, 51, 53, 55, 57, 59

Points (-2,6), (-1,6), (0,2), (1,0), (2,6)

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End of Lesson Reflection: Write your answers on a paper with your name and pass forward.

How well do you understand the lesson?

1- Nothing 2- Somewhat 3- Half 4- Most 5- Everything

What is one thing you learned today?

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