GENERAL FORM OF THE EQUATION FOR A LINEAR REALTION!

GENERAL FORM OF EQUATION!

Ax+By+C= 0

A is a whole number, and B and C are integers.



SOME RULES IN WRITING  IN STANDARD FORM INTO GENERAL FORM

• If the numberes are infraction, find the LCD first, then multiply the LCD to both sides.
• If it is in standard form, you just simply used the 'dig and transplant' method.
• The A value should always be positive.
• If the numbers has a fraction, multiply the denominator to both sides.

Rewriting an equation in General Form:
1.)   4x+3y=36
4x+3y-36=0

2.)  1/3y=5/6x-1/2
 (1st  step)2  6(1/3y)=(5/6x-1/2)6            (LCD:6... then multiply 6 to both sides!)
(2nd step)  2y=5x-3                                 (the answer in 1st step)           
 (collect all the terms on the left side of the equation)
  (3rd step) 5x-2y-3=0                             (This is the general form of the equation!)

3.) y=-2/3x=4
(1st step) 3(y)=3(-2/3x=4)                   (multiply each side by 3)
(2nd step) 3y=-2x=12                          (the answer in 1st step)
(collect all the terms on the left side of the equation)
(3rd step) 2x+3y-12=0                          (This is the general form of the equation!)

Determining the slope of a line given its equation in general form
3x-2y-16=0

(Rewrite the equation in slpoe- intercept form)

-2y-16=-3x     (1st step)
-2y-16=3x /-2  (2nd step)   (divide each side by -2)
y+8=3/2 (3rd step)
y=3/2 x-8

• The slope of the line is 3/2.

Determining the x and y intercept.

3x+2y-18=0

x-intercept:
( substitute y:0)
3x+2(0)-18=0
(solve for x)
3x=18
(divide both sides by 3)
3x/3 =18/3
x=6

y-intercept:
(substitute:0)
3(0)=2y-18=0
(solve for y)
2y=18
(divide each sides by 2)
2y/2 =18/2
y=9

General form into a equation.
8x-2y+10=0
(rewrite the equation into slope-intercept form)
-2y=-8x-10
(divide each side by -2)
-2y=-8x-10 /-2
y=4x+5
or
8x+10=2y /2
4x+5=y
y=4x+5

Nerdy Me

Relations and Functions

Relations and Functions

No one did yesterday's blog entry... Here is some definitions.

"A Relation is when our input (independent/x value) produces an output (dependent/y value). Each input must have at least one output to be a relation."

"A Function is a type of relation where one input will produce exactly one output. Therefore, every function is a relation, but remember every relation is not a function."

If you are ever asked, "Are these functions?" and are presented with a bunch of relations, it's comparable to being presented with a basket of fruit, and having to separate all the apples.
^ Disregard this if that was a horrible comparison. (Attempt to make this post relatable? ☑)

** This blog post will not be showing you the V.L.T. or Vertical Line Test. Sorry, not my job.**

Today's Notes:

Relations and functions can be showed as an equation, graph, or a set of ordered pairs.
Example of an Equation: (y=mx+b)
Example of Ordered Pairs: { (3,2) (4,2) (5,2) }
Example of a Graph:
 

Domain and Range

Domain: the set of all x values that are used as input in a relation.

Range: the set of all y values that are produced as output in a relation.

To write down domain or range you can use "Interval Notation" the simpler way. Or the more complex and widely accepted way, "Set Notation".

Writing Down Domain and Range Interval Notation:

Interval Notation

This is an example graph that we did in class.

To write the domain in Interval Notation for this graph, you simply find the the first and last number on the x axis that this line would sit below or hover over because we have a line between two points, meaning the input could be any number in between these points. If you squint a little, and take a few steps back, you might see that this line(the side with the dot) begins under the -3 on the x axis. This is the first possible input that can be used in the function. The arrow on the other end means that the line continues forever, meaning that the other side of the x axis is extended infinitely. This means the last number on the x axis could be any number above -3, and we'll call this infinity ∞. Therefore to write this in interval form, you'd write:

[-3, ∞)
Congratulations. You did it.
To get your range, you take the line and compare it to it's position along the y axis. Your range is [-2,∞)

But wait, this is the easy way out. The way you should learn this stuff is much more difficult.

(If you come across a graph with points plotted, like the ones in this ordered pair for example, { (3,2) (4,2) (5,2) }

the x values are the first numbers in the pair, and the y values are the ones second.)

Domain: [3,4,5]

Range: [2] or perhaps just 2 is acceptable. I'd mark your page correct either way.

Set Notation

Set Notation is written in this format:
{x/xER}

Translation:

The set of all values of x such that x is an element of the real number system.
*The colour system should be self explanatory*

This means every possible input is part of the real number system.

After this, you write a weird form of interval notation. Using the example above, in the domain, the first number was -3, and the last number was
∞.
Or in other symbols:

x≥-3
Translation:

x is greater than or equal to -3.

All there is left to it, is to tack on the second part, and add it to the first.
{x/xER,x≥-3}
and there is your extremely hard to explain, underwhelming explanation for writing domain.

I won't bother to explain how you get the range for the above example, but here is the Set Notation:

{y/yER,y≥-2}

Apply what you know about domain to range, and there you have it.

Names and or Meanings of the "other symbols"

{ } - Set notation brackets.
[ ] - I call 'em square brackets. They mean "up to and including this number."

( ) - Round brackets? Perhaps? They mean "up to and not including this number." You use these for infinity because you can't quite reach infinity.

Closing

Did I succeed in creating a blog post that was quick, easy to comprehend, and wasn't a wall of text? ☒ Probably not, but I hope it covered everything. So, I know the question we're all wondering here. What if a graph has a line with an arrow on each end? How many numbers will the graph include? Let's ask a mathlete.

(I don't know how terribly relevant this is, but I heard "limit" in class once.)

Exponent Laws

As you all know, we went back in time a few days ago, back... to grade 9. For those of you who weren't there, here is the review on Exponent Laws (Power Laws) .

Rule #1 When multiplying with exponents, simply add the exponents and the equation will become much simpler.

for example:
                     3         8    <--- they're exponents okay?
                ( x     ) ( x    )
                     
               Since both constants in the equation are the same, you can add the exponents.

                    3    +  5          8
                ( x   ) ( x   ) = ( x   )

Of course, you could always write out the equation like this:

                    3       5                                                8
                ( x   ) ( x   ) = (x*x*x) + (x*x*x*x*x) = x

But. People who do math professionally (or anyone who has done it frequently) will know that we are incredibly lazy when it comes to work and always take the shortest route to whatever we are trying to do, so use the first equation (this also applies to all exponent laws that will appear on this post).


Rule #2

When dividing with exponents that have the same constant, simply subtract the exponents from each other.

ex.       5     3        2                                                                                                        
         x    / x    =  x                                                                                                                          
                                                                                                                                                           

Rule #3   

If a constant ever has an exponent of  "0", then the product will always be one.

ex.         0
           x    = 1


Rule #4
                                                                                        5   3    
When a power is the base of another power, like this:  ( x      )     multiply the two exponents together to get one exponent in order to simplify the equation. so it'll look like this:          15
                                                                                                        ( x      )                              

Rule #5

When dividing an exponent by an exponent with an exponent base outside the exponent equation  
                       m   t
 like this:    (  x  )          
                 ____           multiply the inner exponents by the outer exponent (multiply m + n by t).
                       n                
                 ( y     )

Rule #6

If you have a power with a negative exponent it can be written as the reciprocal with a positive exponent.
                            -3          
In other words:   3      =    _1_
                                          3

If I have left anything out from the lesson plans, feel free to comment, or tell me in class when you see me.

Fractional Exponents

This class we learned about fractional exponents.

What fractional exponents do is give us another method of writing radicals.

The mathimatical way of saying it is to take the denominator of the fraction

and use it as the index of the radical, like this:

 271/3 = 3√27

All you do is take The 3 from the 1/3 and put it in front of the radical,

which it is now called the index.

271/ = 3√27

Now the one that is left over you put behind the 27, so what you're 

doing is just grouping the 27 and 1 together, and putting it in the radical,

and they are both called the radicand.

=3√271

Because 271 equals 27, and mathematicians are lazy, if something times itself 

equals the same number, you don't have to put it.  

=3√27

Silly Math Jokes!! :P

JOKE NO.1:

Teacher: "Who can tell me what 7 times 6 is?"
Student: "It's 42!"
Teacher: "Very good! - And who can tell me what 6 times 7 is?"
Same student: "It's 24!"

JOKE NO.2:

Q: What does a mathematician present to his fiancée when he wants to propose?
A: A polynomial ring!



( It's not really related to what we're learning right now..but it's still about math!!)

Radical Rules

Radical Rules

√36 = 6

Try to find out if the radicand has any perfect squares within it. (like 4 and 9).. when multiplied together equal, for example, 36.

√(4)(9)
√4 * √9

 Since 4 and 9 are perfect squares, you can easily find out what number times itself equals them. ( 2*2= 4, 3*3= 9).

2*3 = 6

So, √36 = 6.   Or √4/9 = √4 / √9 = 2/3

Mixed Radical

Example:
√80 
Find the factors of 80. Factor Tree:                      

After, you are left with 2* 2* 2* 2 * 5. 

= √(2x2)(2x2)x5
Two groups of 2's. So 4^2 * 5.
= √4^2 * √5= 4√5Simplify:√63= √9 * 7 = √9 * √7 ^^^ Perfect square! (9)= 3√7                  √63 is between perfect squares √49 and √64. The estimate will be closer around 7.9, since is closer to √64. 
√49 = 7
√63 = 7.9 (guess) ... 7.937253933193772 (actually answer)√64 = 8
Radicals are the opposite of powers.

Mixed Radicals...again.

a√x .. The 'a' value has been simplified or removed from the radical.

Entire         <--->                Mixed√12                                       2√3√32                                       4√2 ....
ex: √32 = √16 * 2 = √16 * √ 2 = 4√2 .. (4x4=16)
or: √32
      = √8 * √4
      = √2 * √4 * 2 = 4√2

Simplify:

√500                      √125                      √96                      √200                       √90                 √112
√100 * √5             √25 * √5               √16 * √6                √100 * √2                  √9 * √10         √16 * √7
=10√5                   = 5√5                  = 4√6                   = 10√2                      = 3√10            = 4√7

Polynomial/Factoring Review Pt2

Backward

Finding factors

ex.

 12- 1*12,  2*6,  3*8,  3*4

24- 1*24,  2*12,  3*8, 4*6

x²- (x)(x)

---

Prime Factorization

ex.          64   

                               

                   8 * 8                                       

                                                           

         2*2*2*2*2*2   

           

---

Reverse Distributive Property

Find the GCF (Greatest common factor)

ex. -64x²+112x

16x(-4x+7)


What did I do? Well lets go forward.

16x(-4x)+ 16x(7)

First -64x² Last 112x

Together it equals  -64x²+112x

Still confused? (I'm not a teacher so yes you are)

---

                                                                                  

Reverse FOIL (refer to page 1 for FOIL)

This may be rough but for a question like this you need to find two numbers when multiplied equal the last term and add together to equal the second term.

So you then  separate the question into two binomials like so.

(x+3) (x+5) You can see I took x² separated it and put the 3 on one side and the 5 on the other. Try FOIL to check your answer.

Please click this link he explains it perfectly 
LINK---------
Reverse FOIL
LINK---------

It will lead you to this video I timed it for this specific point but feel free to watch the entire video.

Alright guys im heading off to the grad awards I will be sure to add more this evening when I return (3 step proccess)

---

The 3 step plan

Alright guys I was going to write out my notes but it seems the wonderful internet has done it for me, click the link below for a detailed explanation on how to factor a polynomial with the three step plan.

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

-LINK-

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Hope this helped.

To students in New York good luck on the test ;)

New York people                                                                Us

Below I will include a link to the first page if you missed part 1 going forward 




Going Forward (part 1)

Polynomial/Factoring Review

Forward

How do I....

   (Monomial)(Monomial)

Solve the question in this order.

1. Sign
2. Coefficant
3. Variable

Ex (-3x²)(7x⁴y)

 We know it will be negative -
multiply Coefficants -21
add variables/exponents -21x⁶y

How do I....

(Monomial)(Polynomial)

Use the Distributive Property

Ex  3(X+7)

=3x+21

We multiply the term in front of the brackets times the terms within them and add the results.

As you can see above -7x² is multiplied through the term and we get our answer below.

How do I....

(Binomial)(Binomial)

Use the FOIL Technique.

(Credit to Free math Help)

How do I....

(Binomial)(Polynomial)

Use the Distributive Property

I recommend checking this site out -LINK- In there page they have examples and interactive tutorials on how to do all these equations. 

Below is a link to part 2 Going backwards

Going backwards (part 2)

Friday, September, 21, 2012.

Natural Numbers - counting numbers.

        ex; 1, 2, 3, 4, 5, 6, 7, 8, 9,10...

            Does not include - zero, negatives, decimals, fractions.

Whole Number Set - 0 plus natural numbers.

        ex; 0, 1, 2, 3, 4, 5, 6, 7, 8, 9...

            Does not include - negatives, decimals, fractions.

Integers - positive and negatives - whole numbers.

        ex; ...-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5...

            Does not include - decimals. 

Rational Set of Numbers - any number that can be written as a terminating or repeating decimal.

Irrational Set of Numbers - can not be written as a terminating or repeating number.

        ex; Pi π

(Baskets example goes here)

(Chocolate bars example goes here)

But Wait ... There's More!

Even - any number evenly divisible by 2.

        ex; 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20...

Odd - any number that can not be divide by 2.

        ex; 1, 3, 5, 7, 9, 11, 13, 15, 17, 19...

Factors - are the numbers that multiply together to produce a product.

        ex; what are the factors of 12. 

                             1*12, 2*6, 3*4 are all factors of 12.

Prime Factors - When we find the factors of a number so that all of those factors are prime numbers.

        ex; find the prime factors of 125 - 5*5*5.

What are the factors of the factors of the following numbers.

1. 144 - 1*144, 2*72, 3*48, 4*36, 6*24, 8*18, 12*12.

2. 225 - 1*225, 3* 75, 5*45, 9*25, 15*15.

3. 81 - 1*81, 3*27, 9*9.

Don't forget about the quiz on Tuesday

Review on Monday

:)

Thought this belonged here. :)