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.