Don't Forget the Properties!

Distributive Property- a (b+c) = ab+bc
Inverse Property- a+ (-a) = 0 and -a + a = 0
Identity Property-a+ 0=0 + a = a
Commutative Property- a + b = b + a and ab = ba
Associative Property- a + (b+c) = (a+b) + c and a(bc) = (ab)c
Multiplication Property of 0- a * 0 = 0 and 0*a=0

Old Vocab Review

Here is some vocab from chapter 2 that you probably need to know for the final :

Type of Linear Equation           Number of Solutions               Indication When Solving
Conditional                                   one                                             final line is x= a number

Contradicition                               none                                          final line is false


Identity                                          infinite solution set                      final line is true, 0=0

Form for joint variation

Some people had trouble with this on the test:
Y varies jointly as x and z:
Y/xz=k if finding the constant is required.

Solving for b in slope intercept form.

It's been a while since we've done this so here is a refresher:

• Find the equation of the straight line that has slope m = 4
  and passes through the point (–1, –6).

Okay, they've given me the value of the slope; in this case, m = 4. Also, in giving me a point on the line, they have given me an x-value and a y-value for this line: x = –1 and y = –6.

In the slope-intercept form of a straight line, I have y, m, x, and b. So the only thing I don't have so far is a value for is b (which gives me the y-intercept). Then all I need to do is plug in what they gave me for the slope and the x and y from this particular point, and then solve for b:

y = mx + b
(–6) = (4)(–1) + b
–6 = –4 + b
–2 = b

Then the line equation must be "y = 4x – 2"




Taken from here: http://www.purplemath.com/modules/strtlneq.htm

Dividing inequalities by negative numbers

When dividing an inequality by a negative number always flip the sign. A more in depth explanation is linked.

http://mathforum.org/library/drmath/view/53287.html

Switching the sign for Inequalities

Whenever x is not on the ride side after you solve an equation, you always flip the sign so that the number will be on the right side and the variable on the left side

Switching the sign for Inequalities

Whenever x is not on the ride side after you solve an equation, you always flip the sign so that the number will be on the right side and the variable on the left side

Percentage Problems

Some people in my class were having trouble with percentage problems. Here is what you need to remember the proportion x/100 = is/of. x = the percentage.

In vs. Exlusive

Don't forget that in notations etc. to check to see id it is inclusive or exlusive. Like for Interval notation if the number line went from -3 (included) to infinty (to the right (positive)) you would want to remember that WHENEVER you see an infinity it is EXlusive. ALWAYS.
So it would end up being [-3,infinty)
Brigid

Set interval and such

Hey guys, so I remembered that there was a little confusion about set and interval notation.
Set: -3Interval: (-3,5)
Brigid

Helpful website

This website has some walkthroughs of some of the mixture problems on the selected problems from chapter 2 google doc. Check the comments, windows won't let me paste it.

2.3 !!!!

I have the same question as christine about rate problems can someone please show me how to do one step by step? Please!

Another Question!

Also, I am a bit confused about the mixture, rate/time, and interest/principle problems.

If someone could go over it, that would be really helpful! :)

Help Please!

Hey everyone!

So looking over my previous test, I was a bit confused about some things! If someone could help me that would be great! :)

I don't understand how to graph

f(x)=3x! or g(x)=1-x^2

I was also confused on how to graph a relation!

remember chapter 3

We only did section 3.1-3.3

DO NOT DO ANY EXTRA SECTIONS TO THROW YOU OFF!!!!

 GOOD LUCK EVERYONE!

Solving compound inequalities with and

Step 1. Solve each inequality in the compound individually.

Step 2.  Since the inequalities are joined with and, the solution set of the compound inequality will include all numbers that satisfy both inequalities in step 1 (the intersection of the solution sets).

Chapter three info

A linear inequality in one variable can be written as Ax + B < C.  This is true when A, B, and C are not equal to 0.

Addition property of inequality
A<B and A+C< B+ C are equivalent.   Adding the same number to each side of an inequality does not change the solution set.

Multiplication property of inequality
for all real numbers A, B, and C, with C not equal to 0
A<B and AC<BC are equivalent if C>0
Here are the steps to solving a linear inequality.

1. Simplify each side separately. Use the distributive property to clear parentheses and combine like terms as needed.

2. Isolate the variable terms on one side.  Use the addition property of inequality to get all terms with variables on one side of the inequality and all numbers on the other side.

3.  Isolate the variable.  Use the multiplication property of inequality to change the inequality to the form of x<k or x> k

Khan academy

On youtube there is a man whose channel is called khanacademy.  He has really helpful videos for math that would benefit everybody in the class.  Here is the link to a video about solving inequalities.

http://www.youtube.com/user/khanacademy#p/c/7AF1C14AF1B05894/4/VgDe_D8ojxw

Remember

When a number is included on a number line, use a closed circle and you have a solid line.  When a number is not included, you use an open circle, and have a dashed line.

The questions we went over in class today

Today in class we went over some of the questions that were asked during class today. We did talk about them but maybe this way you can use Noa's method (putting the equation on one side...) I will put the questions in the comment section of this post and make another with the answers so you can also quiz yourself.
Good luck tomorrow!

Extra Help chapter 1

Natural numbers (counting numbers) = {1,2,3,4,5,6.....}
Whole numbers = {0,1,2,3,4,5,6...} They are counting numbers but they include 0.
Integers = {..., -3, -2, -1, 0 , 1,2,3...}
Rational numbers = {p/q, p and q are integers not equal to 0} 4/1, 1.3, -9/2, 16/8
Irrational numbers = {real number that is not rational.  Square root of 3, negative square root of 2, Pi
Real numbers = a point on a number line.

Study Problems for Chapter 2

Here are some study problems I am using for chapter 2
Reply and comment

Study Problems for chapter 3

Here are some Problems from chapter 3 that I am using to study
I will post the problems and then we can all post the answers to the different problems as comments and that way we are colabrating!

Study Problems from chapter 4

Here are some problems I am doing to study for the book
They cover all the concepts that we went over in class
I hope that they are helpful for you!
They are problems from the book but they are the most important ones for me to consentrate on and study.
The answers can be found in the back of the book.
You can reply on the blog as a comment to answer the questions
that way we are colabarating and helping each other to study and learn! :)
Amelia

Scribe post 12/14 Patrick

Today in class we went over the google document that is meant to help us prepare for finals. Jojo answered questions that people had about it and went over problems. I will do a few of them now:

Rational numbers are numbers that terminate. Irrational numbers do not terminate or repeat. An example of a rational number is 2/3 or 1/3. In decimal form 1/3 is .3333(repeating). An example of an irrational number is pi or 3.1415... pi goes on forever and does not stop.

Absolute value equations:

|2x-1|=7

2x-1=7 -(2x-1)=7

+1 +1 -2x+1=7

2x=8 -1 -1

x=4 -2x=6

x=-3

x=4 or x=-3

A number (v) plus 15% is 287.5

(1)v+.15(v)=287.5

1.15v=287.5

___ ____

287.5 287.5

v=250

250x.15=37.5

37.5+250=287.5

I guess this is the last scribe post of the term.

Important thing to remember for finals! 2.1

When solving an equation, if both sides have the same thing.
Example: x+1=x+1 then the answer is all real numbers
but if both sides are totally different
Example; 2=5 Then the answer is no solution.
Amelia

4.4 and 4.5 Key Terms!

List of key terms you should know before our test!

4.4

1) Linear Inequality in two variables

2) Boundary Line

4.5

1) Dependent Variable

2) Independent Variable

3) Relation

4) Function

5) Domain

6) Range

7) Function Notation

8) Linear Function

9) Constant Function

4.1, 4.2

4.1

1)Ordered Pair

2)Origin

3)X-axis

4)Y-axis

5)Plot

6)Components

7)Coordinate

8)Quadrant

9)Graph of an equation

10)First-degree equation

11)Linear equation in 2 variables

12)X-intercept

13)Y-intercept

4.2

14)Rise

15)Run

16)Slope

Helpful website for "Percent of" word problems.

I was having some trouble with percent problems earlier and this website really helped me. A few of the examples that helped were:

• What percent of 20 is 30?

We have the original number (20) and the comparative number (30). The unknown in this problem is the rate or percentage. Since the statement is "(thirty) is (some percentage) of (twenty)", then the variable stands for the percentage, and the equation is:

30 = (x)(20)

30 ÷ 20 = x = 1.5

And:

If you need to find 16% of 1400, you first convert the percentage "16%" to its decimal form; namely, the number "0.16". (When you are doing actual math, you need to use actual numbers. Always convert the percentages to decimals!) Then, since "sixteen percent OF fourteen hundred" tells you to multiply the 0.16 and the 1400, you get: (0.16)(1400) = 224. This says that 224 is sixteen percent of 1400.

And:

• What is 35% of 80?

Here we have the rate (35%) and the original number (80); the unknown is the comparative number which constitutes 35% of 80. Since the exercise statement is "(some number) is (thirty-five percent) of (eighty)", then the variable stands for a number and the equation is:

x = (0.35)(80)

x = 28

Twenty-eight is 35% of 80.

Here's the link: http://www.purplemath.com/modules/percntof.htm

Info from sections 3.1, 4.1, and 4.4

Here is some other info from :

3.1

( ) are for less than or greater than problems.

[ ] are for less than or equal too, or greater than or equal too problems.

4.1

Vertical line has an undefined slope.

Horizontal line has a slope of 0

Lines that start from the bottom left corner and go up on the graph have a positive slope.

Lines that start form the top left corner and go down have a negative slope.

4.4

Shading above or below the line on inequalities:

A good way to check to make sure you shaded in the right area is by plugging in a point from the shaded area and if it makes the inequality true than the area is correctly shaded.

Something to remember is if the inequality is less than or greater than then the line is dotted. And if the inequality is less than or equal too or greater than or equal too, then the line is solid.

Good Ways to Study

1. Google Docs

2. Test Aftermath

3. Blog

4. Youtube Videos!

5. Cumulative Review in Book!

4.4 Linear Inequalities Notes

Here is the google doc for 4.4.
It has some really good examples and answers some questions you might have.

How To Study For Finals!

http://www.teencollegeeducation.org/2010/01/crunch-time-study-tips-how-to-study-for.html

This is a great site if you need any tips on how to study for Finals!

Good Luck!

Some helpful google documents

Because its sometimes really difficult to remember terms (especially when its from the first couple of chapters we went over,) I will put the links to some helpful links in the comments. Hope it helps!

Chapter 1 notes

The link to the chapter 1 tutorial are in the comments section of this post!

Chapter 2 Tutorial

So even though we only made tutorials for the first few chapters I still think they would be really helpful to look over, they have key terms, example problems and so on. It won't let me paste the link for some strage reason but if you just go to your google docs it will be right there. I will aslo try to attach it in the comments section. Good luck studying!

Good things to do while taking the final

Here are some helpful things to do while taking the final to make sure you do your best!

1. Always check your work whenever you can to make sure your answer is right!

2. Take your time and don't rush through problems because you may make a careless mistake!

3. Always read every problem carefully, especially word problems, because it is important to find out all the information that is given, and what you are supposed find in the problem. (There are a lot of word problems in chapter 2 that I think are hard, so practicing a lot of those will help)!

4. Read all of the directions given throughout the test because you don't want to skip over something accidentally!

5. Always try each problem, even if you don't know what to do! It is always better to try because you may get some credit, even if the answer is wrong!

How to Study Math!

Another way to study math, that I have found useful, is to write a sample problem, on a flashcard, of the math you are working on,and study / memorize it! I have found this useful for learning how to do word problems. Another way to memorize how to do a word problem, is to write the “equation”, or how to do it on a piece of paper 10 times...This is also a very good way to learn!

Hope this helps!! Good luck on the Final!

Studying For Final Exam

Here are some ways to study for the final exam!
Go through your old tests and mark areas that you did poorly on and practice re doing the problems.
Amelia

Chapter 4 Review Post! Graphs Linear Equations, and Functions

Here is the chapter 4 review post to help for our final test!

Chapter 3 Review Post! Linear Equations and Absolute Value

Here is the Chapter 3 review post for finals!

Chapter 2 Review Post! Linear Equations and Applications

Ok so this post can be a chapter 2 review post for finals! Hope it helps! :)

Chapter 1 Review!

Ok so this post can be a chapter 1 review post! Just comment on it to add things from chapter 1! This will really help everyone study for finals.

Final Review: Chapter 1, Review of the Real Number System

1.1: Basic Concepts

-Natural Numbers

otherwise known as counting numbers {1,2,3,4,5,6,...}

-Whole Numbers

{0,1,2,3,4,5,6,...}

-Integers

{...,-3,-2,-1,0,1,2,3,...}

-Rational Numbers

{p/q⎮p and q are integers, q≠0}

Examples:

-Irrational Numbers

{x⎮x is a real number that is not rational}

Examples: √x, -√2, pi.

-Real Numbers

{x⎮x is represented by a point on a number line}

-Additive inverse

-Number Line

-Set Builder Notation

More Variations

The area of a circle varies directly as the square of the radius= A=kr^2

Pressure varies inversely as volume. P=k/v

For a given principal, intrest varies jointly as rate and time: I=krt

Variation

y=kx^n, then y varies directly as or is proportional to, x^n

y= k/n then y varies inversely as x^n

y=kxz,then y varies jointly as x and z

Function notation practice problem

2x+3y=12 subtract 2x

y=-2/3x +4 divide by 3

f(x)= -2/3x +4

Video Explanation of Function Notation

This lady explains function notation and domain and range pretty clearly. I hope the video in some way helps yall too.

http://www.brightstorm.com/math/algebra/graphs-and-functions/function-notation

Practice Problems - Variation

After looking over the first and some of the second page these practice problems were pretty helpful.
This website covered all the types of variation equations

http://www.purplemath.com/modules/variatn.htm

Differences between Domain and Range and Independent and Dependent Variables

Domain- Set of values for the indepedent variable (x)
Range- Set of values for the independent variable (y)

Indepedent Variable- Experimental variable (what you manipulate)
Dependent Variable- What is affected during the expermiment

Vertical Line Test, and Horizontal Line test

Vertical Line Test- The relation represents a function when every vertical line intersects the graph of a relation in no more than one point


Horizontal Line Test- A graph passes the horizontal line test when its not possible to draw a horizontal line that intersects the graph in two or more places

Jared Scribe Post

Today in class JoJo taught us how to solve a joint variation problem and an inverse variation problem.  When solving an inverse problem the equation to solve is: y= k/x.  Once you find K, the constant, you can solve the problem.  To solve a joint variation problem you have to use the equation y=kxz.  Y varies jointly as x and z if there exists a real number k.  I was confused on how to solve these problems before today, but after the lesson I felt very confident with solving joint and inverse variation problems.  The rest of the class time was used to review for the test!  The next scribe will be done by Cole

 GOOD LUCK EVERYONE!

Test review

Harrison said...
slope intersept form: slope 3/5; y-intercept (0,-8)
y=mx+b y=3/5x+-8
This is helpful to remember while graphing inequalities.
Also remember: y=x - solid line w/ no shadeing
y>x - dashed line, all above shaded
y y>_ - solid line, all above shaded
y<_ - solid line, all under shaded
{(-4,2),(-4,-2),(1,5),(1,-5)}
Domain={-4,1} Range={2,-2,5,-5} this is not a function.

Varies directly y=kx
Varies inversly y=k/x y decreases as x increases
Varies jointly y=kxz - the two variables are x and y
Combined variation y=kw/h
Remember: k is a constant, it will never change

Domain: any value of x variables
Range: any value of y variables
A relation represents a function only if all x values are unique.
while graphing a relation, plot the points, don't connect them.
f(x)= just a formal name for Y

OK - lots of fun stuff to help you on the test, I'm sure most of what I just said has already been posted but this is my summery of things you may want to look at before tomorrow.

Helpful ways to review!

Here are some things that I am doing to review for this test, I hope you find this helpful! One thing is make sure to remember that with you are graphing a line with a '>/<' sign, that the line will be dotted, and when you are graphing a line with a less than or equal to/greater than or equal to sign that the line is solid. Also practicing shading the different sides is very helpful for section 4.4, on pages 206-208 in our books have good practice problems on the right and left sides, and the problems have the answers at the bottom (they have these problems in all of the sections). In section 4.5 make sure to practice using the vertical line test and giving the domain and ranges in functions. Also make sure to remember that f(x) is the same thing as y. In section 4.6 practice all types on variations (direct, inverse, joint, combine) and make sure that you know how to set up all of the different types. On page 242 in our books there is a summary of lessons 4.4-4.6 and it explains the key points that we learned about and gives examples. I found this page super helpful, so make sure to look at it!

Harrison said...
Here's how to solve a inverse variation problem.
If z varies indersely as w, and z = 10 when w = .5 find z when w = 8.

z=k/w
10=k/.5
multiply by .5
k=5

z=5/8 z=.625

Definition of a Relation

Ist variable paired with one or more values of the second variable. (x.y)

(1,3), (1,5)

Ex. The change you recieve for buying a $.65 soda depends on how much you gave the vending machine. x .65 .70 .75 1.00 5.00

y 0 .05 .10 .35 4.35

Domain Restrictions

Values of the variable that will make the function undefined, values not included in the domain.

y=1/x D: All real numbers when x isn't equal to zero. R: All real numbers when y is greater than or equal to zero or when y is greater than 0.

Linear Functions

Multiply by the same umber over and over again.

y=output or dependent variable

x=input or independent variable

Things to remember

In order to have the variation equation you must solve for k.
Amelia

Things to remember

In order to have the variation equation you must solve for k.
Amelia

Scribe Post WITH IKE! AAAHHH YAY 4.6

OK! So, get pumped. Because 4.6 is really just... a great chapter. Really, it is the BEES KNEES.

What we covered today in this chapter of wonder is only the first part of it: direct variations.

According to our trusty book, these types of equations are extremely common in everyday business and physical science life. If those are part of your life, this may mean the world to you.

When Y is said to 'VARY DIRECTLY AS X', it means that Y is dependent on a multiple of X.

This is known as direct variation. An easy way to remember this is to just remember the formula

Y = KX .

Good? What this means is basically what was just said; that Y is dependent to a multiple of X: aka, KX.

(y is proportional to x.)

So, in order to be able to say this, really powerful people who can make laws about math have decided to give the number that you multiply x with (k) a handy dandy name. They call it...

THE CONSTANT OF VARIATION!

Whew.

In a direct variation, (which is what were covering right now,) as long as K is greater than zero...

AS X INCREASES, Y ALSO INCREASES.

So, in a real world situation, K would be the constant in the problem. For instance, if you want to find the cost for buying yummy in my tummy candy bars, (Snickers, of course,) they will always cost the same. I mean, theoretically, and for the sake of the problem, the cost will never change. This means that the cost is CONSTANT, meaning it is THE CONSTANT OF VARIATION, or K! Easy enough. The equation to write this would be:

y = (COST OF SNICKERS) x

Y would equal your final cost, and x would be how many snickers you bought. And, again, since the cost of the snickers is positive (k), the more snickers you buy, the higher the cost will be.

There!

And that is about it for DIRECT VARIATION problems, in simplicity.

JARED is the next scribe.

Direct variation and Joint variation

Direct variation: Y varies directly as x if there exists some constant k such that y=kx

joint variation: Y varies jointly as x and z if there exists a real number k such that y= kxz

Inverse Variation

y varies inversely as x if there exists a real number k such that: y= k/x

also y varies inversely as the nth power of x if there exists a real number k such that y=k/x^

Solving a Variation Problem

Step 1= Write the variation problem
Step 2= Substitute the initial values and solve for k.
Step 3= Rewrite the variation equation with the value of k from step 2.
Step 4= Substitute the remaining values solve for the unknown and find the required answer.
Amelia

Function Machine!!

http://www.amblesideprimary.com/ambleweb/mentalmaths/functionmachines.html

This is the website to the function machine we played with earlier in the week... maybe it will help for our next test?!?

Examples of Function Notation, and Definition of Relation!

Definition:

A relation is a correspondence between two sets (called the domainand the range) such that to each element of the domain, there is assigned one or more elements of the range.

FUNCTION NOTATION

We write f (x) to mean the function whose input is x

Example

If

f(x) = 7

then

f(12) = 7

Here f is called the constant function. Whatever comes in to f, the number 7 comes out.

http://www.ltcconline.net/greenl/courses/152a/functgraph/relfun.htm

This is a great website if you need example problems, or help on the upcoming test!

function notation

y=f(x). This is function notation.

To solve an expression for f(x) step 1: solve the equation for y.  Step 2: replace y with f(x).

Here is an equation
Find F(-2)
F(x) = X squared +1
F(-2) = (-2) squared +1
F(-2)= 4+1
F(-2) = 5

Reminder about Relations

Remember that a relation is not always a function because a relation only represents a function if all x values are different. So a relation can or cannot be a function, it all depends on the x values. When you graph a relation that is not a function, you don't connect the dots on the graph, you just leave them as points rather than a line since it is not a function.

Domain and Range

Domain= all the possible values of x.
Range= all the possible values of y.
Amelia Hess

function

An easy way to tell if a graph is a function or not is to use the vertical and/or horizontal line test.  If the vertical or horizontal line passes through the graph more than once, than it's not a function.

Example of functions

X=0,1,2,1,0
y=3,4,5,6,7
this function doesn't work because the same x has two different x's have the same y.
X=1,2,3,4,5
y=1,2,3,4,5
this works and is a function because none of the x values have the same y.
Amelia

Functions

For every x there is only one y.
Amelia

How you write a function

Y =3x+2 is how you write a linear equation
You write the same thing with an f(x) for it to be a function
F(x)=3x+2
f(1)=3(1)+2
f(-2)=3(-2)+2
Amelia

Function

If any two points on the graph have the same x value, it is not a function. Vertical lines are not a function.
Amelia

Function

Christine, 4.5 introduction to Functions

Today in class, we started learning about functions! At first, we all thought that it would be super confusing! But when looking into the chapter, it makes a lot of sense. We use a lot of information we learned in past chapters. We did a digital scavenger hunt on google docs which helped us understand functions. I feel like I am starting to really understand math and especially functions. I thought that it was really cool that x and y values are also known as domain and range. At first, I really didn't understand the what a function was, but the definition in the book made it very clear to me. A function is a relation in which, for each value of the first component of the ordered pairs, there is exactly one value of the second component.

Sorry the pictures of graphs wouldn't upload but I will post them later!

When Graphing a System

It may be helpful to shade with a different color for each inequality you graph.

class today

Today in class we went over how to graph linear inequalities.  REMEMBER, when you see <  or > to have a solid line and a close circle.  When you have < or > there is an open circle which means the line is dotted.

harrison said

In class today people were a little befuddled as to what the x and y intercepts were. The x intrcept is where the line of your graph crosses the x axis and the y intercept is where the line of your graph crosses the y axis. To find the x intercept let the y value equil 0. To find the y intercept let the x value equil 0.

Steps to solve a linear inequality

Step 1: Solve for Y

Step 2: Find X and Y intercepts

Place a graph using open or closed circles and be sure to label your graph!

Step 3: Determine proper line depending on inequality (solid or dotted line) and connect intercepts.

Step 4: Determine half plane shading

Check: Plug in ordered pair into the inequality and see if the statement makes sense

System of Linear Inequalities

We want to know all the ordered pairs that satisfy both inequalites.
We can find this by plotting the lines on the graph and shading in the correct spots. The spot where all the shading is the same for the lines contains the points that will satisfy all of the inequalites.
Amelia

Ordered Pair

Ordered Pair= x coordinate that is directly connected with the y coordinate. A line is made up of ordered pairs.
Amelia

Half Plane

Half Plane= The shaded part bellow or above the line. The half bellow the line and the half above the line go on infinately. 50% each.
Amelia

solving linear inequalites

plug in zero for y to find the x intercept.
plug in zero for x to find the y intercept.
Example:
y0x>-2
yy<0+2
Y<2
(-2,0),(0,2)
Amelia

Vocab

x axis= the point where the line crosses the x axis.
y axis= the point where the line crosses the y axis.
I hope you guys find this helpful
Amelia

Vocab List so far...

So today in class (at least in our class) Jojo gave us a list of words that we should know the definition so I will put them here so you will all have them for the test.

X-Axis: The horizontal number line in a rectangular coordinate system, could also mean where the line passes through the horizontal number line.

Shaded Region is the part of the graph that is shaded, based on whether y is greater or less than the equation.

Half Plane : is the part of a plane that lies on one side of the line.

Ordered pairs: are pairs of numbers that you use to locate a point on a coordinate grid. The first number in the pair will tell you how far to move horizontally (X) , and the second number tells you how far to move vertically (Y)

Definition of a Linear equation

A linear equation is when x appears only to the first power, as in the equation below. A linear equation is also called an equation of the first degree. (The degree of any equation is the highest exponent that appears on the unknown number.)

2x+5=27

Solving Linear equations helpful site

Yesterday we got back our 4.1-4.3 test. This website can help serve as a review of solving linear Equations

http://www.purplemath.com/modules/solvelin.htm

4.5 Vertical Line Test

Vertical Line Test
If every vertical line intersects the graph of a relation in no more than one point, then the relation represents a function.
Amelia

Variations of the Definition of Function!

1. A function is a relation in which, for each value of the first component of the ordered pairs , there is exactly one value of the second component.

2. A function is a set of ordered pairs in which no first component is repeated.

3.A function is a rule or correspondence that assigns exactly one range value to each domain!

4.5 introduction vocab

Domain and Range
In relation the set of all values of the independent variable (x) is the Domain.
The set of allvalues of the dependant variable (y) is the range.
Amelia

Key Terms

Relation: Any set of ordered pairs.

Function: A relation in which, for each value of he first component of the ordered pairs, there are exactly one value of the second component!

Other things to remember

Dotted line means not included.
Not dotted line means included.
Amelia

Important things

Shading above the line is on top of the roof
shading bellow the line is inside of the hose.
The shaded area has infinatly many points.
Amelia

Really Helpful Site!

This site is so much fun! It is kind of like gizmo but a little more interactive! Use this if you need help graphing linear inequalities!

http://www.mathwarehouse.com/algebra/linear_equation/linear-inequality.php

Helpful Tricks

There are many helpful tricks and tips when graphing linear inequalities.

I know that I was confused at first when you are supposed to shade in different sides. I always thought that to the left would mean less than, and to the right would mean greater than. But when graphing linear inequalities, use Jojo's trick for remembering. To graph a greater than inequality, shade "on the roof" or the left side. To graph less than inequalities, shade "in the house" or the right side. This method has helped me a ton while graphing linear inequalities!

Remember

Remember you can visit explorelearning.com to help with shading

scribe post 11/22/10

The problems and examples that we go over in our book and in class aren't always all that relevant to our daily lives. We often use inequalities when are just going about our lives. Here is an example that will put this into context. You have $150 in your pocket and you are stopping by a shopping plaza on the way home. Your mom asks you to pick up some groceries while your out. Your mom needs you to get eggs, milk, bread, beef, apples, and gatorade. The eggs will be $6, the milk is $5, the bread is $2, the beef is $8, the apples are $5, and the gatorade is $4. All these groceries together are $30. You also would like to buy a pair of shoes and get some nike socks. If your shoes cost $110. The socks cost $2.50 for one pack with three pairs of socks. How many pairs of socks can you buy? $6+$5+$2+$8+$5+$4+$110+$2.50(x) is less than or equal to $150 you would combine like terms (add together everything except 2.50(x)) and you would get $140+2.50(x) is less than or equal to $150 then you would subtract $140 from both sides and get $10 is less than or equal to 2.50(x) so x is less than or equal to $5 in context this means you can spend up to five dollars on socks. which gives you two packs of socks.

congrats

Congrats to the Varsity Boys Basketball team on beating the Chamblee Bulldogs.  The pythons had a tough match up against a bigger, taller Bulldogs team , but came out with a win at the final seconds!!!!!  The score was 40-39 Paideia!!!      GO PYTHONS!!!

  Jared Maner and Emma Ming Kayhart were there.  

Graphy the Union of two Linear Inequalites.

Or= When two inequalites are joined bu the word or; we must find the union of the graphys of the inequalites. The graph of the union of two inequalities includes all of the points that satisfu either inequality.
Amelia

Graph the Intersection of two Linear Inequalities

And= Graph the Intersection of two Linear Inequalities. A pair of inequalites joined with the word and is interpreted as the intersection of the solution sets of the inequalites. The graph of the intersection of two or more inequalites is the region of the plane where all points satisfy all at the same time.
Amelia

Steps to Graphing a Linear Inequality

1. Draw the graph of the straight line that is the boundary. Make the line solid if the inequality involves greater than or equal to or less than or equal to. Make the line dashed if the inequality involves < or>.
2. Choose a test point. Choose any point not on the line, and substitute the coordinates of this point in the inequality.
3. Shade the appropriate region. Shade the region that includes the test point if it satisfies the original nequality; otherwise,shade the region on the other side of the boundery line.
Amelia

Steps to Graphing a Linear Inequality

Linear Inequality in Two Variables

An inequality that can be written as Ax+ByC
Where A B and C are real numbers and A and B are not both zero, is a linear inequality in two variables.
Amelia

Solving Linear Inequalities

when you are solving a linear inequality you can solve it using y=mx+b but instead of using the equals sign you would use an inequality sign such as y>mx+b or yAmelia

Linear Inequalities

When solving linear inequalities there are some important things to know
Where you shade= for less than bellow the line and for greater than above the line
Sollid Line= When it is equal to or when you use a solid line and shade on one side or the other it is less than or equal to or greater than or equal to.
Dotted line= Less than or greater than and you can shade on either side.
Amelia

How to find B in Y=Mx + B

Well I use point slope form

y - y1 = m(x - x1) this equation is helpful for finding b in y = mx + b

dont forget

you can visit this link to help with slope and more!!!


http://www.youtube.com/user/khanacademy

Slope

This is just an example, but DO NOT forget to line up the Ys and Xs.

For Example as you see above, it is

Y2 - Y1

------------

X2 - X1

Do not forget this!!

Test Taking Skills for Math!

http://www.mathpower.com/tip4.htm

This is a great site if you need any study skills relating to math.

Good Review for the Chapter 4 Test

When I was reading some sections from the book, I found some really helpful pages that would be good to look at when your studying. On page 241 in our book, it summarizes the sections that we learned, and the main points that the sections covered. On the left side of page 241 it gives examples from each section, and it shows you all of the steps for each problem. Also on pages 239 and 240 there are all of the key terms in chapter 4 that we should know about, and some vocabulary questions that you can do to make sure you understand the terms. I hope you guys find this as helpful as I did!

The Reasons Why Slope Intercept Form is Used More!

The slope-intercept form of a linear equation is the most useful and widely used for several reasons. Every linear equation (of a non-vertical line) has a unique slope- intercept form. Linear functions are also defined using slope-intercept form. Also, this form is what we use when graphing a line with a graphing calculator!

Although slope-intercept form is used more, using point-slope form can be extremely helpful when doing problems that ask: Write an equation that goes through the point (7,2) that is parallel to 3x-y=8. It is more helpful this way because you don't have to go through the trouble of finding out what the x and y intercepts are through plugging in for zero!

Equations of Horizontal and Vertical Lines!

A horizontal line through the point (a,b) has equation y=b.

A horizontal line is also known as zero line.

A vertical line through the point (a,b) has equation x=a

A vertical line is also known as an undefined line.

The Many Different Ways to Say Slope

Slope can be defined as:

-rise over run

-the change in y over the change in x

-y2-y1 over x2-x1

-y1-y2 over x1-x2

-the average rate of change

To say "the change in" in mathematical terms, you use the delta sign: ∆.

Finding Intercepts and a Helpful Tip!

When graphing the equation of a line, to find x, plug 0 in for the y-intercept. To find y, plug in 0 for x!

Also a really good tip to know your line is correct is....

While two points, are sufficient to graph a straight line, it is a really good idea to use a third point to stay away from errors!

Remember what the Quadrants are!

What I do to remember this is just remember that the quadrants go counter-clockwise.

Ways to Study for the Test

Here are some ways to study for the test:
1. Do the study problems on the blog
2. Look back at you notes from this chapter and practice important consepts.
3. Go over the problems you got wrong or had trouble with from previous homeworks.
4. Look up problems from this chapter online.
5. Review the group homework on gooogle docs.
Amelia

Definitions of Dependant and Independant

Dependant= A variable whose value depends on the values of one or more independent variables.
Independant= A variable in the equation whose values make up the domain
Amelia

A Meathod We went over in Class

AX+BY=C
-AX -AX What you do to one side of an equation you must do to the other
BY=-AX+C/B
Divide by B on both sides
Y=-A/B+C/B
M=-A/B
B=C/B
I hope you guys find this helpful!
Amelia

Helpful Test Review!

You guys have been doing a great job posting.  As a gift, I have supplied a few helpful review problems for tomorrow's test.  Some of these questions might even show up!




Find the line parallel to y = 3x + 4 that contains the point (-1, 5)
Graph both lines.


Find the line perpendicular to 4x - 3y =12 that contains the point (8, 2).  Graph both lines.

Find the slope of the line that passes through the given points.

 (1, 3) and (2,6) (2, 8) and (2, 6)

                                                                     

The slope a vertical line is _________________________.

The slope of a horizontal line is _____________________.

The ­­­­­­­­­­­­­___________________ is where a function crosses the x-axis.

The ­­­­­­­­­­­­­___________________ is where a function crosses the y-axis.

 

Put each of the following equations into slope-intercept form, y = mx + b.  Identify the slope, m and the y-intercept, b.

y – 2x = 2

y = __________________________  m = ____________________  b = _____________

y = x

 

y = __________________________  m = ____________________  b = _____________

Graph the following equations. Be sure to label the x and y intercepts (if applicable).

y = -2x + 3                                                                    y = -x           

 

 

Caroline has a small business making bath and body beauty baskets.  She estimates that her fixed weekly costs for rent, electricity and salaries is $350.  The products and supplies for one basket cost $5.50.

1. If Caroline makes 20 bath and beauty baskets in a given week, what will her weekly costs be?

2. Caroline’s total costs for last week were $700. How many bath and beauty baskets did she make?

Point- Slope Form vs. Slope-Intercept Form

This is a way easier way to solve certain problems with point-slope form rather than slope-intercept form.

Slope-Intercept Form: y=mx+b, m=slope, b=y-intercept.

Point-Slope Form: (pretend , is sub one!) y-y‚=m(x-x,)

Here is an example of this problem:

Find a line parallel to this that goes through the points (6,-2) 4x-y=3

y-y,=4(x-x,)

(y-(-2))=4(x-6)

y+2=4x-24

y=4x-22

Parallel Slope

The slope of a line parallel to another line is the same slope.
Amelia

Tyler, Scribe Post 11/16/2010

Today in class we started by JoJo handing back DQ#13 and we went over problems 2 and 3 as a class. Discussing ways we could set up problems like that. After that he came around and checked the chapter review which every one had completed. We checked over it then went over problems #16 and #23. JoJo reminded us that slope and the rate of change are the same things. We worked on Point-slope problems here is a example.

y-y=m(x-x)
y-4= -9(x-1)
y-4=-9x+9(add four on both sides)
y=-9x+13

Two different ways to solve for the X and Y intercepts

When solving a problem where you have to find a perpendicular line you must find the x and y intersepts for the perpendicular line.
you plug in the point given to you into either form
y= mx+b
slope intercept form
or
y-y=m(x-x)
point slope form
Amelia

Finding Point Slop Form

Good site!!

Key Terms

Ordered Pair: An ordered pair is a pair of numbers written in parentheses in which the order of the numbers is important.  

Origin: When two number lines intersect at a right angle, the origin is the common point 0.

X-axis: The horizontal number line in a rectangular coordinate system. 

Y-axis: The vertical number line in a rectangular coordinate system.

Rectangular (Cartesian) Coordinate system: Two number lines that intersect at a right angle at their 0 points form a rectangular coordinate system, also called the Cartesian coordinate system.

Plot: To plot an ordered pair is to locate it on a rectangular coordinate system.

Components: The two numbers in an ordered pair

Coordinate: Each number in an ordered pair represents a coordinate of the corresponding point

Quadrant: A quadrant is one of the four regions in the plane determined by a rectangular coordinate system

Graph of an equation: The graph of an equation is the set of points corresponding to all ordered pairs that satisfy the equation

First degree equation: A first degree equation has no term with a variable to a power greater than one.
Linear equation in two variables: A first degree equation with two variables is a linear equation in two variables


X-intercept: Point where a line intersects the x-axis


Y-intercept: Point where a line intersects the y-axis

Rise: vertical change between two points on the line


Run: Horizontal change between two points on the line


Slope: Ratio of the change in y compared to the change in x along a line is the slope of the line

Forms of Linear Equations

Slope Intercept: y = mx + b
The slope and y-int can be easily identified and used to quickly graph the equation.

Point Slope: y - y1 = m(x -x1)
This form is ideal for fing the equation of a line if the slope and a point on the line or two points on the line are known.

Standard: Ax + By = C
The x- and y- ints can be found quickly and used to graph the equation. Slope must be calculated.


Horizontal Line: y = b
If the graph intersects only the y-axis, then y is the only variable in the equation.

Vertical Line: x = a
If the graph intersects only the x-axis, the x is the only variable in the equation.

Scribe post 11/11/10 Patrick

Today we started the class with Jojo talking about scribe posting. If nobody does it after a couple of days it is recommended that you step up and do it. If nobody does it after a while then Jojo will start giving people zeroes. So it is a good idea to take charge of scribe posting before everybody starts getting bad grades. We also learned that we should try to comment as much as you can instead of making new topics. Many of the posts on the blog could be comments on other peoples posts. By doing that it would not take up as much space. It would also save people time because all of the information on a subject would be in one place. Keep the blog clean I guess. Next we talked about our homework on the google document last night. You were required to do four problems but if you did more or want to do more you get extra credit. If you haven't done the homework it is advised that you do it now because there are not many problems left. Do it quick or there will be no more problems. You can also correct peoples work for extra credit. Do not delete what they did, mistakes are as important as correct problems. Just keep what they did and do your work in a different color. Do not show your name on the document. Jojo can tell who did what and when they did it.

The next scribe is Tyler.

Every Triangle

parallel and perpendicular lines

Here is a link to help with parallel and perpendicular lines

http://www.purplemath.com/modules/slope2.htm

Danielle, Website that explains slope with examples

here's a website that can be helpful because it gives you examples on how to figure out slope and it shows you what undefined slope is and also this website has some important definitions that we have learned.

Ten Ways to Survive the Math Blues

The other day Jojo showed our class an article online that I thought was very cool, useful, and helpful. I think it would be good to share with the class again because for people that have trouble in math like me, it's hard to actually even go to class or like math at all. So here are ten ways to survive math because you'll have to take it for the rest of high school!

1)Figure out the big picture

2)Get on top of it before it gets on top of you

3)Read Ahead

4)Use more than one resource

5)Don't join the blame game

6)Practice makes perfect

7)Time management

8)Don't copy to survive

9)Never, never give up

10)Have fun with it!

The most important ones to me are numbers 5-9. NOT that the others aren't cool. But I think it is important that everyone should that math will get hard and super boring and when it does, keep in mind that by doing it more often you eventually master it. All you need is time, a positive attitude, and determination! If you think about it, you feel much better and more successful when you work hard for something and finally get it rather than not work very well and get a crappy outcome. Copying applies to not working very well because you aren't doing your own work. Remember to not blame your teachers because they are only trying as much as you will so give as much as you can and they will too. The very important ones are left for last: don't give up and have fun!

I hope this helps you as much it has helped me. :D

remember

If the slope of two lines are the same, the lines are parallel and if the lines slopes are the negative reciprocals of each other then they are perpendicular.

More About Slopes

As we know, the equation for slope is: m=Y2-Y1/X2-X1, but, this equation can also be written as: Y1-Y2/X1-X2. It doesn't matter which equation you choose to use, but you must be consistent. You are allowed to use both of these equations because they both mean the same thing:Change in Y/Change in X, which is written as: Δy/Δx, because the greek symbol Delta(Δ) means change.

Scribe Post 11/8/10

In class today we went over homework and we also talked about equations. Today we talked about how the point on the X axis is a vertical line and a vertical line has an undefined slope. X is undefined because its a vertical line and anything you cant walk on without falling has an undefined slope.
Ex: x=2







Also we learned that the point on the y axis has a slope of zero.
Ex: y=2














Today we also learned that parallel lines have the same slope and that perpendicular lines have a slopes that are negative reciprocal of each other.

here are some equations that might help you find certain forms:
y-y1 =m(x-x1) is point slope form.
Ax + By = C and thats standard form.
Y= Mx + B is slope intercept form

The scribe poster for tomorrow is patrick

Positive and negative slope

A positive slope means that the line goes up (rises) from left to right.
A negative slope means that the line goes down (falls) from left to right.

chap 4 notes

Perpidicular lines have the same slope

_I_ have slopes that are negative reciprocals of each other

a/b=-b/a

2 negative reciprocal is -1/2

a=-1/a

Also;

Slope is the ratio of rise over run.

Extra Help

Here is a link of a youtube video to help understand slope.  The guy has a whole channel that has plenty of videos for math!

  I hope this helps
 
   http://www.youtube.com/watch?v=hXP1Gv9IMBo

Remember!!

When using slope intercept form, y=mx+b, always remember that y is not the y intercept, it is actually b! Also, remember that m is slope!

Slope and Slop Intercept Form

Slope Equation:

M= Rise/Run = Y2 - Y1 / X2 - X1

Slope Intercept Form =

Y = MX + B

Free graph paper for 4.1

Here is a link for free graph paper for the upcoming lessons

http://incompetech.com/graphpaper/

New Symbols used in Chapter 4

Here are some new symbols that will be used throughout chapter 4. We will learn more about these symbols later on, but hopefully you will get the basic idea of what they mean from this post. 

(a,b)=Ordered Pair

x with a '1' underneath it= A specific value of the variable x(read as "x sub one")

m= Slope

f(x)= Function of x (read as "f of x")

*Sorry I wasn't sure how to make the "x sub one" symbol, but it's like an exponent underneath the 'x' instead of above it. 

Practice

Hey Guys!

If anyone needs help on Graphing Inequalities this is a great site with examples and practice problems!

http://www.math.com/school/subject2/lessons/S2U4L3GL.html

4.1

So far we know how to solve a linear equation in two variables.  Remember that Ax + By= C and this is STANDARD FORM

Inequalities Reminder

Always remember that when you multiply or divid by a negative number, the inequality symbols FLIPS.

Example: -2x>-4

/-2 /-2 = x<2

On the test always make check your answer to make sure you flipped the sign when multiplying or dividing by a negative.

Danielle, Inequalities

here's a link to practice absolute value equations and inequalities

http://www.blogger.com/post-create.g?blogID=2979614811124141785

When One Side of Inequality is Divided by a Value

Ex.
6a + 3 < -3
   -4
First multiply both sides by -4

6a + 3(-4) < -3(-4)
6a + 3 > 12

Now subtract 3 from both sides to isolate the coefficient 
6a - 3 > 12 -3
6a > 9

Now divide 6 from both sides to isolate the variable
6a > 9
6      6
a > 2/3

graph!


--(--------------------->
 2/3