Domain and Range

The domain of an absolute value function is (-inf.,inf) and its range is [0,inf.)

The reciprocal function the domain and range are both (-inf.,0) or (0,inf.)

The square root function the domain is [0,inf.) and the range is [0,inf.)

Every Transformation

F(X)= -afunc(x+/- h)+/-K

(reflection (shift left (Shift up or

over x-axis) or right down same as

opp. sign) sign)

Every Transformation

Every Transformation

-the standard form for any function is:
f(x)=-a(x-h)+k

f(x)= - a func (x +/- h) +/- k

- =reflection

a= stretches/flattens

+/- h= shifts left or right

+/- k= shifts up or down.

f(x)=2x^2+3, parabola y=x^2

q(x)= 4x^3, cubic, y=x^3

g(x)=1/(x-3), allometric, 1/x=y

w(x)=x+27 -6, square root, yx

h(x)=-(x-3), absolute value, y=x

Types of Functions and what they're called.

f(x)=2x^2 + 3, parabola ,y=x^2
g(x)= 4x^3,cubic ,y=x^3
g(x)=1(x-3)=allometric, 1x=y
w(x)= (x+27)-6, square root ,y=x
h(x)= -x-3 ,abs. value, y=x

Wednesday, March 30, 2011

Hey, I found a website with images of graphed functions with all different degrees.
Also, remember the graph names for functions with x^2, Parabolic
And the function with 1 over X is Allometric
You guys should take a look, just click it, they're only graphs, hehe, no reading.[kinda]
http://library.thinkquest.org/2647/algebra/functype.htm

New Vocab

Abscissa = The X coordinate. The distance from a point to the vertical or y axis measured parallel to the horizontal or X axis.
Ordinate = nThe perpindicular distance of p from the x axis. The values of x and y together written x, y.

Vertical Line Test

Vertical line test if a vertical line passes through more than one point of the graph. then the relation is not a function. A parabola is an example of a function because if you draw a line through the graph it doesnt go through more than one point.

Re-Introduction to Functions Presentation

Re-Introduction to Functions

 


Assignment 1 Homework Links


1.  http://www.intmath.com/functions-and-graphs/1-introduction-to-functions.php


2.  http://www.intmath.com/functions-and-graphs/2b-functions-from-verbal-statements.php


Assignment 2 Homework Links


1.  http://www.intmath.com/functions-and-graphs/3-rectangular-coordinates.php

example

f(x) = 4x + 7x, g(x) = 7y + 2x

f(10) = 4(10) + 7(10), f(10) = 40 + 70, f(10) = 110

g(7) = 7(7) + 2(10) = 49 + 20 = 69

HW

what were the links for the homework?

Today we talked about real life examples of functions, so here's a few to help you guys grasp the concept.

The amount of air pollution is dependent on the number of cars on the road.

The distance a baseball is hit depends on the force it is struck with.

A person's hunger is dependent on the the amount of food they have recently consumed.

( time is always the independent variable because nothing can control time.)

Function notes

A function is a rule that relates how one quantity depends on other quantitites

V=IR where

V=Voltage (V)

I=current (A)

R= resistance

s=speed (m/s)

t=time take (s)

If d increases the speed goes up


S=1mi/t t is independent

y=x^2 + 3 time is on the x- axis and independent variable

independent varible represents all possible values of domain

function f of x...f(x)

Monday, March 28, 2011

Remember to NOT MULTIPLY!
These mathematical statements all mean the same!
Linear Equation:
y = 2x +3

Linear Function:
f(x) = 2x + 3

g(x) = 2x + 3

h(a) = 2a + 3

Chap 8. Rational Expressions

A rational number is the quotient of two integers, with the denominator not 0. In algebra a rational expression is the quotient of two polynomials with the denominator 0

x/y -a/y m+4/m-2 8x^2-2x+5/4x^2+5x

Just vocab...Expanding is the same thing as Foiling.

7.3 Special Factoring

Difference of Squares

x^2-y^2=(x+y)(x-y)


Perfect Square Trinomials
x^2+2xy+y^2=(x+y)^2
x^2-2xy+y^2=(x-y)^2

Difference of Cubes
x^3-y^3=(x-y)(x^2+xy+y^2)

7.2 Factoring Trinomials

Choose factors of the first term and factors of the last term. The place them in a pair of parenthesis

( )( )

Use different combinations of the factors until the correct middle term is found

Factoring out a Binomial factor

(x-5)(x+6)+(x-5)(2x+5)
the greatest common factor is (x-5)
(x-5)[(x+6)+(2x+5)]
commutative prop. of +
(x-5)(x+6+2x+5)
combine like terms
(x-5)(3x+11)

and that's how you factor out a binomial factor

More on X Method

When the polynomials first term is greater than one this is how you factor it:
You do all of the same steps as the first example.
1. Then when you rewrite the answer, you put the first term in the original problem, and divide it by the second term after you factored it.
2. Then you simplify the problem.
3. Next, if you have a fraction, then you take the denominator and multiply it by the first term(in the factored out problem)
4. That is your answer.

Factoring X Method

For factoring when the first term is 1:
You put the first and last term in the bottom of X and multiply them. You put the middle term in the top of the x. You find two numbers that add to get the middle term, and multiply to get the first and last term multiplied together. You put those numbers in the two sides of the x. Then you rewrite the new problem and that is the answer.

Factoring a Polynomial

These steps are listed int he book on page 393 and they help so remember to use these on the test tomorrow

Step1 Factor out any common factor
Step 2 If the polynomial is a binomial, check to see if it is the difference of sqaures
Step 3 If the polynomial is a trinomial , check to see if it is a perfect sqaure trinomial
Step 4 If the polynomial has more than 3 terms, try to factor the grouping
Final Step- Check the factored form by multiplying

Don't forget...

Don't forget to look over some word problems, 2nd period went over one in class and it seems like they could get a little confusing. some practice problems are on 403. remember to keep track of all of your variables and details!
good luck tomorrow!

Extra help

hey!so as I am studying for our test tomorrow I found that it is really helpful to do the chapter 7 test, it has a few different variations of problems that we didn't see on the review.

It also has a lot of factoring by grouping which we worked a lot in class.

Reaaaaally Easy Way to Factor

While I was studying, I was a little bit confused on how to use the AC Method. I found this amazing site that does the method a little differently but makes it so much easier!

http://people.richland.edu/james/misc/acmeth.html

chap 7 review nots

AC METHOD

6X^2-14X+9X-21 6x^2-5x-21 expanding (3x-7)(2x+3) factoring
2X(3X-7)+3(X-7)
(3x-7)(2x+3)


-126
/ \
-14 9

Factoring a Polynomial

Step 1: Factor out any common factors.

Step 2: If the polynomial is a binomial, check to see if there is a difference of squares.
If the polynomial is a trinomial, check to see if its a perfect square trinomial. If its not factor it using the methods we have learned.
If the polynomial has more than three terms, try to factor by grouping.

Step 3: Check your answer

Types of Factoring

Types of factors you can use on the test:
Factoring by Grouping
Guess and Check
Ac Method
X-Method
Box Method

Factoring a polynomial

Steps for factoring:

1) Factor out GCF

2) Check to see if the polynomial is a difference of squares

3) If the polynomial is a trinomial check to see if it's a perfect square trinomial.

4) If the polynomial has more than 3 terms try to factor by grouping

5) Check the factored form by multiplying

Solving Quadratic Equations by Factoring

Here are the steps to solving quadratic equations
 #1 Write in standard form.
Standard form= ax^2+bx+c=0
a is not equal to 0
#2  Factor the polynomial
#3 Use the zero-factor property. Set each variable factor eqaul to 0
#4 Find the soultion or solutions- Solve each equation
#5 Check each solution in the original equation

purple math is always god site to go to if you still have trouble
http://www.purplemath.com/modules/solvquad.htm

Special Types of Factoring

You need to know these special types of factoring for the test on Tuesday

Difference of Squares- x^2-y^2=(x+y)(x-y)
Perfect Sqaures Trinomial- x^2+2xy+y^2=(x+y)^2  or  x^2-2xy+y^2=(x-y)^2
Difference of Cubes- x^3-y^3=(x-y)(x^2+xy+y^2)
Sum of Cubes- x^3+y^3=(x+y)(x^2-xy+y^2)

How to factor a perfect Square trinomial

Just to clarify on how to factor a perfect square trinomial:

You take the square root of a term (the first term in the expression) and the square root of the c term (the last term) and multiply them together and then multiply what you get by two. If that is the middle term then you have a perfect square trinomial.

a^2 -(2)(a)(c) + c^2

(a-c)^2

Zero prop. product

Pq=0, then p=0 or q=0
Factor out Gcf
(x-3)(x+6)=0
P Q

x-3=0 or x=6=0 x=-6 x2+5x+6=0
+3 +3 (x+2)(x+3)=0 x+2=0 or x+3=0
x=3 x=-2 x=-3

Special Types of Factoring

Difference of Squares:
x2-y2=(x+y)(x-y)

Perfect Square Trinomial:
x2+2xy+y2=(x+y)^2

Difference of Cubes:
x3-y3=(x-y)(x2+xy+y2)

Sum of Cubes:
x3-y3=(x-y)(x2-xy+y2)

Chapter 7 Review

Here are a list of the things that are covered in chapter 7:

-Greatest Common Factors and Factoring by Grouping (7.1)

- Factoring Trinomials (using all different methods, 7.2)

-Special Factoring (7.3)

- Solving Equations by Factoring(7.4, I'm not sure if we will cover this before the test or not though)!

If you go to page 405 in our books there are some multiple choice questions that ask about different terms which is one really helpful way to study. The answers to these questions are on the next page at the bottom!

WEDNESDAY, MARCH 16, 2011

I found this problem a little confusing on the class/home work today but, with some help, was able to figure it out.
4z^2+4zw+w^2 factored the problem looks like...
(2z+w)(2z+w) simplify to look like...
(2z+w)^2
I realize that this is not a difficult problem, but I thought the problem was not a perfect of squares.

CHAPTER 7 TEST DAY -MARCH 22

CHAPTER 7 TEST DAY -MARCH 22

FACTORING IS THE MAJOR TOPIC!  Get Ready to do well...

The X-Method

The X-Method:
I hope this helps! I know it may be kind of hard to understand on here, but I hopefully you will get it!

Ex: 6x^2z^2+5xz-4

Step 1: Draw an X and put the middle term (B term) in the top of the X and the first and last terms multiplied in the bottom of the X (AC)






Step 2: Think of two factors that add to get 5, and multiply to get -24.
The two factors are 8 and -3, so you put 8 and -3 on the two sides of the X.

Step 3: Now re-write the new equation.
(xz+8)(xz-3)

Step 4: Use the ‘bottoms up’ method to find the answer. First step to the bottoms up method is to put the A term as the denominator below the 8 and -3.
(xz+8/6)(xz-3/6)
Now you simplify this:
= (xz+4/3)(xz-1/2)
The next step of the bottoms up method is to multiply the denominators by xz, so you basically move the bottom of the fraction up!
= (3xz+4)(2xz-1) This is your answer!

Step 5: Foil to check!
(3xz+4)(2xz-1)= 6x^2z^2-3xz+8xz-4= 6x^2z^2+5xz-4 This is the right answer!

Factoring a Trinomial Using the AC Method

Today we learned about SPECIAL CASES, the two kinds of special cases are...
1. x^2+2xy+y^2 factored this problem will always look like (x+y)(x+y), the problem starts and ends with a perfect square.
2. x^2-y^2 factored the problem looks like (x+y)(x-y)
COOL

Today in class I was asked to explain a home work problem to the class, here are some terms that I should have used:
4x^2-4x-15
4x=the leading coefficient
-15=the constant

Guess and Check method for factoring trinomials

Homework Problems!

Here are some homework problems that we went over in class that many people found difficult!

I am using the guess and check method!

33. -15a^2-70a+120

1. Find the GCF for polynomial

-5(3a^2+14a-24) -24

6,-4

-5(a+6)(3a-4)

35. -11x^3+110x^2-264x

1. Find the GCF for polynomial

-11x(x^2+99x^2-11)

-11(x-6)(x-4)

40. 4(m-5)^2-5(m-5)-15 (m-5)=x

4x^2-4x-15

(2x+3)(2x-5)

(2(m-5)+3)(2(m-5)-5)

(2m-10+3)(2m-10-5)

(2m-7)(2m-15)

The Box Method -Explained by Paideia Students

6k^2 - 19k + 10 is the polynomial they are factoring!

Also to help clairify:
Standard Quadratic Method- the set up looks like ax^2+bx+c
Just incase anyone missed that in class.
TANKS!!!

Helpful tip for factoring.
2y^3+8y^2-10y we can make this problem have aleading coeficient of one by simply extracting 2y from the first term.
2y(y^2+4y-5) now just factor
2y(y-5)(y-1) thats all, word up!!!

Box Method

Heres a helpful link for how too do the box method

http://seattlecentral.edu/faculty/alevy/Box_%20Method.pdf

Box Method

The Box Method is also known as the Abc method because a*c= b and a+c=b . Creating a 2 by 2 box lays out the all the terms you are working with so you can get a better picture visually of how to factor.

The AC method for factoring.

I found this great video that explains how to solve using the AC method.

Factoring

Standard Quadratic Form

ax^2+bx+cx^0


Strategies-
Factor Out GCF
Factor By Grouping
Guess and when a=1

x^2+1x-12
(x+4)(x-3)

x^2-2x-24
(x-6)(x+4)


2x^2+x-6
x(x+3)(x-2)

Factoring a Trinomial with a Common Factor

Factor 16y^3+24y^2-16y
16y^3+24y^2-16y=8y(2y^2+3y-2)
= 8y(2y-1)(y+2) GCF=8y. Remember the common factor

Factoring Trinomials

1st step to factoring trinomials: Choose factors of the first term and factors of the last term.
2nd Step: Next place the factors in a pair of parentheses like this (       )(       )
To check if you have the right combination you use the FOIL method

Example Problems:

x^2- 8x+15              4x^2-8x-21
(x-5 )( x-3 )               (2x + 3 )(2x - 7)
x*x= x^2                  2x*2x=4x^2
-5*-3= -15              2x*-7=-14x
-3+-5= -8                -14x+6x=-8x
                                 3*-7=-21

factoring by grouping

5x^2
//// 25                           5x times x              2x^2+6x
///                                                  2x(x+3)
5x5
25x1

GCF: largest value that can go into each term of an expression or set of numbers

3x(4x-5)+5(4x-5)
(4x-5) gcf

ax^2+bx+c

a: leading coefficent
b: middle term coefficent
c:constant

look for integers (r and s) such that rxs=c and r+s+b so that the factored expression looks like such
(x+r)(x+s)

(3x+2)(x-4)-(3x+2)(x+8) this is a binomial not a polynmial.
(3x+2)(x-4)-(x+8)simlify by foiling
(3x+2)(-12)distribute the negative sign and combine the last two terms
-12(3x+2) fully factored!

Factoring Trinomials - YourTeacher.com - Algebra Help

If you need help with Factoring Trinomials i hope this well help

Factoring trinomials

p^2+5p+6

ax+bx+c

r*s2+3= (p+3)(p+2)
p^2r+2p+6

Factoring Trinomials

P^2+5P+6 Guess and check

(p+3)(p+2)

p^2+3p+2p+6

p^2+5p+6 r x s/2+3

someone can comment and explain the steps