Polynomial+Functions+-+Lesson+Summaries

September 14: What do you Remember?

updated by: Gillian Evans
__Factoring Review:__ Factoring is a tool used in mathematics to break functions down into their various components so when they are multiplied together they result in the original expression or function. In order to be successful in this course, you will need to recall how to factor various bionomials and trinomials. Here's a handy list of the types of factoring you will use regularly:

__Characteristics of a Function?__ 1) General Overview: http://www.purplemath.com/modules/fcns.htm 2) Terminology a) End Behaviour - What happens to the y-values as x approaches positive or negative infinity b) Symmetry - even = line symmetry about the y-axis odd = point symmetry about the origin (can be rotated 180 degrees to conicide with itself) c) Quadrants - Functions extend from one quadrant to another (read left to right) d) Domain/Range - set of all x/y values e) Interval Notation - 3 methods = number line, inequality, brackets "(" up to and not including, "[" including

__Homework:__ Complete both of the following: Bring this one to class to hand-in:

updated by: Shagufta Panchbhaya
__Characteristics of Polynomial Functions:__ - variable - term(s) - exponent on the variable - terms connected by +/- - can be graphed - can be put in decreasing orders of x - can be coefficients and/or constant terms - exponents are positive and whole numbers

__Examples of Polynomial Functions:__ f(x) = x^2 + 2x -1 f(x) = __3__x^4+ x^3 - 2 f(x) = x^5+ x^4 + x^3+ x^2+ x+ 1 f(x) = __-__x^7 + 2x^2 f(x) = x^2 + 2x -1 f(x) = __1/2__ x^2 + 3/2x -1/5 f(x) = 7

- the **degree** is the highest exponent in the polynomial function. For example, in the function f(x) = __3__x^4+ x^3 - 2, the degree is 4. - the **leading coefficient** is the number before the x. For example, in the function f(x) = __3__x^4+ x^3 - 2, the leading coefficient is 3. __Things We Know__ 1) A negative leading coefficient means the function ends in quadrant 4 and vice versa. 2) If degree of function is odd, the function starts and ends at opposite sides of the x-axis. Refer to: [] for greater detail

__Examples__

1) Which are polynomial functions? Why or why not? g(x) = sinx * is not a polynomial function  f(x) = 2x^4 k(x) = x^3 - 5x^2 + 6x -8 m(x) = 3^x *is not a polynomial function because x is the exponent

2) Come up with examples of polynomial functions whose graph extends from:

a) Quadrant 3 to 1: y = 3x^3 b) Quadrant 2 to 4: y = 3(-x)^3 c) Quadrant 3 to 4: y = -3x^2 d) Quadrant 2 to 1: y = 4x^2

References - handout **Power Functions and End Behaviour of Power Functions** - []

__Homework:__ Complete pg 4 # 5-7, 14 in textbook

Refer to my scanned notes if needed: [|MathNote.jpg]

updated by: Shagufta Panchbhaya
__Finite Differences:__ For a polynomial function of degree n, where n is a positive integer (ex: f(x) = x^n), then nth differences:

1) are equal

2) have the same sign as the leading coefficient

3) the value of the nth differences is equal to: (leadingcoefficient)(n!)

__Factorials:__ - indicated by a ! - n! means the product of all the numbers from n to 1 Ex: 5! = 5 x 4 x 3 x 2 x 1 = 120 Ex: 26! = 26 x 25 x 24 x 23 x 22 ...... x 1

__Example:__ Find the degree of the polynomial, the sign, and value of hte leading coefficient.

degree --> 3 (the third finite differences are all the same, or constant: 6) sign --> + (the third finite differences are positive) leading coefficient --> 1
 * = x ||= y ||= FIRST ||= SECOND ||= THIRD ||
 * = -3 ||= -36 ||=  ||=   ||=   ||
 * = -2 ||= -12 ||= 24 ||=  ||=   ||
 * = -1 ||= -2 ||= 10 ||= -14 ||=  ||
 * = 0 ||= 0 ||= 2 ||= -8 ||= 6 ||
 * = 1 ||= 0 ||= 0 ||= -2 ||= 6 ||
 * = 2 ||= 4 ||= 4 ||= 4 ||= 6 ||
 * = 3 ||= 18 ||= 14 ||= 10 ||= 6 ||
 * = 4 ||= 48 ||= 40 ||= 26 ||= 6 ||

(leadingcoefficient)(n!) = constant value (LC)(3!) = 6 LC = 6/3! LC = 6/6 LC = 1
 * To calculate the leading coefficient:**

When calculating finite differences remember: number under - number above

__References:__ Handout : **What is the relationship between finite differences and the equation of a polynomial function?** Handout: **Problem** (using Excel and regression) For more information about calculating finite differences: []

__Homework:__ Handout: **Key Features of Graphs of Polynomials** Handout: **1.7.2: Factoring in our Graphs**

Refer to my scanned notes if needed: [|MathNote!.jpg]

=September 20: Graphing Polynomial Functions= Updated by Wade Walker

We can use the factored form of a polynomial function to graph it. The key information to use is: 1. Where are the x- intercepts? ex. f(x)= x(x-2)(x=3) x=0, 2, -3

2.What does the leading coefficient tell us? If it is positive the functions ends in quadrant 1 (the functions ends up). If it is negative the functions ends in quadrant 4 (the function ends down).

3. What exponent is on each factor? If the exponent is 1 the functions passes through the x-intercept, if it is 2 the function touches then "bounces off" the x-axis and if it is 3 the function acts like a cubic ( "spends more time"). ex.  f(x)=(x-1)(x+4) 2 (x-6) 3

4. Even/ Odd Check. If the functions has an even degree it will start and end on the same side of the x-axis. If the function has an odd degree it will start and end on opposite sides of the x-axis.


 * Homework**: Handouts: "What's my Polynomial Name?", "Factoring Polynomials- Finding Zeroes of Polynomials", and "Unit #1:Polynomial Functions".

Additional references:http://www.youtube.com/watch?v=PmBNhKRhpqE&feature=related http://zonalandeducation.com/mmts/functionInstitute/polynomialFunctions/graphs/polynomialFunctionGraphs.html

September 21: Even and odd revisited
Updated by Joanne Kim

Terms from "1.7.3: What's My Polynomial Name?" homework sheet: - inflection point = turning point - multiplicity = degree of factor

__Finding the leading coefficient given one point__ Given point (2,2) on a curve, facing down, going through -2, bouncing off 1, and through 3. f(x) = -a (x+2) (x-1)^2 (x-3)

Sub in the x and y value for (2,2): 2 = -a (2+2) (2-1)^2 (2-3) = -a (4) (1) (-1) = 4a a = 1/2

cautionary tale: a might not always be the value of 1 *check for the leading coefficient if given a point.

1) Even functions **: f(x) = f(-x)**
 * __Even and odd revisited__**

Ex: Is g(x) even? g(x) = x^2 - 8 g(-x) = (-x)^2 - 8 = x^2 - 8 Therefore, yes, g(x) is an even function.

2) Odd functions **: f(x) = - f(-x)**


 * Ex: If f(x) odd?**
 * f(x) = x^3 - 4x**
 * - f(-x) = -[(-x)^3 - 4(-x)]**
 * = -(-x^3 + 4x)**
 * = x^3 - 4x**
 * Therefore, yes, f(x) is an odd function.**


 * More examples:**
 * f(x) = 2x^4 - 5x^2 + 4 (Even)**
 * f(x) = -3x^5 + 9x^3 + 2x (Odd)**

f(x) = -x^4 + x^2 f(x) = -cos x || f(x) = sin x f(x) = x^3 f(x) = -3x^5 + x^3 ||
 * Even Functions || Odd Functions ||
 * f(x) = |x|


 * Can a function be both even and odd?**
 * Ex) y=0**
 * This is a rare case, though, and once determined even or odd, it's not necessary to disprove the opposite.**


 * Can a function be neither?**
 * Yes, if it does not have even or odd symmetry, it will be neither.**


 * For more information, visit:**
 * http://www.purplemath.com/modules/fcnnot3.htm**


 * For additional practice, visit:**
 * http://www.analyzemath.com/function/even_odd.html**


 * __Homework:__**
 * p 3 # 1-3, 13, 15**
 * p 8 # 1-6, 9**
 * p 12 #1-5, 7, 8**
 * p 16 #4, 5, 8**

September 22: Quest!
===updated by: Patrick Lasagna ===


 * Not much to say is there? We had a Quest today, it covered everything we learned up until this point.**

Friday, September 24, 2010: Dividing Polynomial Functions Updated by: Eva Klimova

Sorry OSCSS Math students, my word document would not copy and paste properly onto this page, so here is the document link. Hope it helps! I pasted two links just in case one does not work.



Monday, September 27, 2010:The Factor Theorem Updated by: Clinton D'Silva

So similar to Eva I can't copy and paste my word document so heres the link!! Have fun



HOMEWORK

pg 35 # 2-4, 9-12

Overview of THE FACTOR THEOREM []

Need Help with Synthetic Division? []

__Tuesday, September 28, 2010: Factor Difference + sum of cubes__ Updated by: Joanne Kim

Factor: x^3 - y^3, where x is a variable, and y is a random number standing for something. Check f(y) : (y)^3 - y^3 = 0 therefore, (x-y) is a factor.

Dividing out (x-y) __y__| 1 0 0 -y^3 __y y^2 y^3__ 1 y y^2 0

therefore, x^2 + xy + y^2 remains.


 * Factoring the difference of cubes: x^3 - y^3 = (x-y) (x^2 + xy + y^2)**

Ex) x^3 - 125 = x^3 - 5^3 = (x-5) (x^2+5x+25)

Factor x^3 + y^3 Check f(-y) : (-y)^3 + y^3 = 0 therefore, (x+y) is a factor.

Dividing out (x+y) __-y |__ 1 0 0 y^3 __-y y^2 -y^3__ 1 -y y^2 0

therefore, x^2 - xy + y^2 remains.

Ex) x^3 + 64 = x^3 + 4^3 = (m+4) (m^2 - 4m + 16)
 * Factoring the sum of cubes: x^3 + y^3 = (x+y) (x^2 - xy + y^2)**

Teamwork Tuesdays! Q: Find a polynomial that passes through points (-2, -1) (-1, 7) (2, -5) and (3, -1)

Homework: Polynomial review, Factoring sum and difference of cubes.

Extra information: []

by:Federico Palacios
 * __Wednesday September 29 2010__**
 * __Solving Polynomials Equations!!__**

Solving polynomial equations = finding their factors! (x intercepts)

Ex) **Solve!** a) x^3 - x^2 - 2x = 0 x (x^2 - x - 2) = 0


 * x (x - 2)(x + 1) = 0**
 * x = 0, 2 and -1**

b) 3x^3 + x^2 - 12x - 4 = 0

factors of 4 = __+__ 4, __+__ 2, __+__ 1 f (2) = 3(2)^3 + (2)^2 - 12(2) - 4 f (2) = 0 -> therefore (x-2) is a factor!

__2 |__ 3 1 -12 -4 __6+ 14+ 4+__ 3 7 2 0

3x^2 + 7x + 2 3x^2 + 6x + x + 2 3x (x +2) + 1 (x + 2) See exercises relating to silos and whales :)
 * f(x) = (3x + 1)(x + 2)(x - 2)**
 * x = -1/3 and __+__ 2**
 * What if our polynomial is __NOT__ factorable????**

__**Friday, October 1st, 2010**__ : **Solving Polynomial Inequalities** By: Eva Klimova





__Homework__: Solving Polynomial Inequalities Assignment **DUE** Tuesday, October 5, 2010.

updated by Sarah Drury
 * Tuesday October 5 2010**

Today we had our Polynomial Functions unit test!

The Teamwork Tuesday task for this week is: Conjugate Roots (Teams of 3) - In class to be presented on chart paper Two roots of a quartic equation are complex conjugates of each other. What can this quartic equation be? What are its roots? Look up any words you don't understand :) []