1. Circle – locus of a point

2. Parabola – locus of a point and line

3. Ellipse – sum from foci is a constant → E: c² = major² – minor²

4. Hyperbola – difference from foci is a constant → H: c² = a² + b²

Formulas:

1. Circle ↔ (x – h)² + (y – k)² = r²

2. Parabola ↔ (x – h)² = 4p(y – k)

3. Ellipse ↔ (x – h)²/a²  + (y – k)²/b² = 1

4. Hyperbola ↔ (x – h)²/ a² - (y – k)² /b² = 1

Examples: Which conic is which? (Coach Higgy’s Easy version)

1.) 4x² - 9x + y – 5 = 0     → Parabola:     (only one thing is squared)

2.) 4x² - y² + 8x – 6y = 0      →Hyperbola:     (- sign)

3.) 2x² + 4y² – 4x +12y = 0     →Ellipse:     (+sign w/ 2 different #’s)

4.) 2x² + 2y² – 8x + 12y + 2 = 0     →Circle:     (+ sign w/ 2 same #’s)

General Equation: (Honors Algebra  2 Book’s confusing way) < Just in case you were interested…

Ax² + Cy²+ Dx + Ey + f = 0

Circle:     A = C          A ≠ 0

Parabola:     AC = 0          A = 0  or C = 0 , not both

Ellipse:     AC > 0          A + C  have like signs

Hyperbola:     AC < 0         A + C  have unlike signs

Examples: Classify      *fish out x² and y²*

^ like this guy!!! haha

1.) x² – 6x + 9 = -y² – 4y

                               +y² +4y

x² + y² – 6x + 4y + 9 = 0     → Circle (+ sign, 2 same #’s)

2.) 4x² = y² +4x + 3

                 -y²

4x² - y² = 4x + 3     →Hyperbola ( – sign)

3.) 9x² + 8y = -228 – 4y² + 90x

                                        +4y²

9x² + 4y² = -228 + 90x     → Ellipse ( + sign, 2 diff. #’s)

4.) -4x² + y²  -8x + 6y – 4 = 0

       -4x² + y²     → Hyperbola (- sign)

Finding Standard Form:

1.) x² + 8x – 8y + 16 = 0     →Parabola (x² = 4py)

          *seperate x and y’s.

   x² + 8x ___ = 8y -16___

          *add C + complete square

   8/2 = 4² → 16 = c

          *fill in the blanks with the C’s!

   x² + 8x + 16 = 8y – 16 + 16

       = (x + 4)² = 8y

2.) x² + y² – 10x + 2y – 74 = 0     →Circle (x² + y² = r²)

          *seperate x and y’s, leave blanks for C and d

(x² - 10x + C) + (y² + 2y = d) = 74 + c + d

          * find c and d

   -10/2 = -5² → 25 = c         2/2 = 1² → 1 = d

           *fill in C and d

   (x² – 10x + 25) + (y² + 2y + 1) = 74 + 25 + 1

   = ( x – 5)² + (y + 1)² = 100

3.) 9x² + y² + 72x -2y + 136 = 0     → Ellipse

              * seperate x and y’s, leave blanks for C and d

(9x² +72x + C) + (y² – 2y + d) = -136

              * factor out 9, complete the square for C and d

 8/2 = 4² → 16 = C          -2/2 = 1² → 1 = d

              *fill in the C and d

9(x² + 8x + 16 ) + (y²- 2y + 1) = -136 + 1 + 144

9(x + 4)² + (y – 1)² = 9

                *divide everything by 9 in order to get 1 on the right side

 =(x + 4)² + (y – 1)² = 1

          1                   9

4.) y² – 4x² – 18y – 8x + 76 = 0     →Hyperbola

                *seperate x and y’s.

 ( y² – 18y) – (4x² – 8x) = -76

                 *leave blanks for C and d, factor out 4

(y² – 18y + C) – 4(x² – 2x + d) = -76 + c + d

                  *complete the squares

-18/2 = -9² → 81 = C         -2/2 = -1² → 1 = d

                   *fill in the blank C and d

(y² – 18y + 81)  – 4(x² – 2x + 1) = -76 + 81 + 1

(y² – 18y + 81) – 4(x² – 2x + 1) = 6

                     *factor down , divide everything by 6

(y – 9)² /6   – 4(x – 1)² /6 = 1

 = (y – 3 )²  – 2(x – 1)²= 1

           2                    3

And now for the best part………………

HOMEWORK!!!!!, haha not! 

H.W. p. 628: 4 – 11, (13 – 19) odd,  29 – 38 , 51- 56

ONLY 7 DAYS LEFT UNTIL SUMMER BREAK!!!!!!!!!!!!!

The standard equation for an ellipse is:

(x-h)²/(a)² + (y-k)²/(b)² = 1

a = the horizontal from center to a vertex
b = vertical from center to a vertex
*Center is at (h,k)

So, if the center is at (0,0), the equation will be:

(x²)/(a²) + (y²)/(b²) = 1

Finding the Foci

the distance to foci from center is distance C, where:

c² = major² – minor²

Example – (x²)/(9) + (y²)/(36) = 1

1. center = (0,0) <– because there is no h or k
2. vertex = (0, ±6) <– found by b
3. covertex = (±3, 0) <– found by a
4. focus 1 = (0, √27)
5. focus 2 = (0, -√27)

1.  5/x=7/9     cross multiply

45=7x

x=45/7

2.  5/x+2=8/x+1     cross multiply

5(x-1)=8(x+2)

5x-5=8x+16

-5x-16=-5x+16

-21=3x

x=-7

More complicated problems

1.  4/x+5/2=-11/x      multiply the top by x so then the x’s on the bottom will cancel out

4+5x/2=-11

then times the top by 2 so the 2 on the bottom will cancel out

8+5x=-22

-8           -8

5x=-30

x=-6

2.  5/x-2=7+10/x-2     times the top top x-2 so the x-2 on the bottom will cancel out

5=7x-14+10

5=7x+14

-5         -5         

7x=9

x=9/7

so when do you know when to cross multiply and when not to???

cross multiply when a/b=c/d ac=bc

do not cross multiply when a/b=c/d+e

More problems

1.  4x+1/x+1=12/x^2-1+3       the LCD is (x+1)(x-1) so times the top by (x+1)(x-1)

(4x+1)(x-1)=12+3(x+1)(x-1)

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

4x^2-3x-1=12+3x^2-3

x^2-3x-10=0

(x-5)(x+2)=0

x=5, x=-2

2.  2/x^2-x=1/x-1     cross multiply

2(x-1)=x^2-x

2x-2=x^2-x

0=x^2-3x+2

0=(x^2-2)(x-1)

x=2, x=1

3.  8/x+2+8/2=5      multiply the top by 2 and by (x+2)

16+8x+16=10x+20

32+8x=10x+20

-20 -8x   -8x-20

12=2x

x=6

4.  -2/x-1=x-8/x+1     cross multiply

-2(x+1)=(x-8)(x-1)

-2x-2=x^2-9x+8

0=x^2-7x+10

0=(x-2)(x-5)

x=2, x=5

HOMEWORK!! p. 571   5-10, 15-18, 21-26, 33-35

x+3/5 + 7/5 = x+10/5

^{_{4/3x – 3/3x= 1/3x}}

_{^{5/x + 7/y, 5*y/x*y + 7*x/y*x = 5y +7x/xy}}

_{^{Now move to some more complicated ones:}}

^{_{1. 5/6x + x/12x,  5*2x}3 /_6*2x3} + x/12x⁵ = 10x^{3 _{+ x/ 12}5
(factor out an x)}
= x(10x² + 1)/ x* 12x⁴ = 10x² + 1/12x⁴

^{2. 5/4x + x/4x^3 + 12x = 5/4x + x/4x(x^2 -3) =5*(x^2 -3)/ 4x(x^2 -3) + x+ 4x(x^2-3)=5*(x^2-3) + x/4x(x^2-3) = 5x^2 + x-15/4x(x^2-3) ( keep the bottom demonimator the same)}

^{You always want to find the LCD: Least Common Denominator}

^{15/5x+1 +16/5x, 15*x/(5x +1)*x
LCD: 5x(x+1)}

^{1/x(x-6) + 12/x^2-3x-18, 1/x(x-6) + 12/(x-6)(x+3)
LCD: x(x-6)(x+3)}

^Examples:

^{1. 4x/(x+1) + x/(x+1)-4/x,  x*4x/x*(x+1) + x*5/x(x+1) – 4*(x+1)/ x*(x+1),  4x^2 + 5x -4(x+1/x(x+1)= 4x^2 + 5x -4x-4/x(x+1)= 4x^2+x-4/x(x+1)}

^{2. x+1/x^2 +4x+4 – 2/x^2-4, (x+1)*(x-2)/(x-2)*(x+2)(x+2) – 2/(x+2)(x-2)*(x+2), x^2-x-2-2x-4/(x+2)(x-2)(x+2) = x^2-3x-6/(x+2)(x+2)(x-2)}

^{Homework: pg. 565 5-7, 12-23, 29-31}

(5xy^3)/(20x^3y) which simplifies to:

y^2/4x^2

and

(x^2+5x)/x^2 which simplifies to:

x(x+5)/x^2 which simplifies further to:

x+5/x

We know how to simplify each quickly.

Now, simplify the rational expression:

((5x^2y)/(2x^3))*((6x^3 y^2)/(10y)) which simplifies to:

(3x^5 y^3)/(2xy^4) which simplifies further to:

3x^4/2y

Now for some more complicated rational expressions…

*****While doing this, remember that (a-b)/(b-a)=-1*****

x^2+3x+2/(x+2)(1-x)

***Now factor the top***

(x+2)(x-1)/(x+2)(1-x) which simplifies further to:

(x-1)/(1-x) *or* (x-1)/(-x+1)

Next, try this one out…

(x-2)(x+3)(x-5)/(x+3)(x-2)(x+5) which simplifies to:

x-5/x+5

And now, try this one…

(4x-4x^2 / x^2+2x-3) * (x^2+x-6 / 4x) which we then factor into:

((4x-4x^2 / (x-1)(x+3)) * ((x+3)(x-2)/4x)) which then turns into:

4x(1-x)(x-2) / (x-1)*4x which simplifies to:

-(x-2)

And finally, try this one…

(5x/3x-12) / (x^2-2x/x^2-6x+8) which we then turn into:

(5x / 3x-12) * (x^2-6x+8 / x^2-2) which we make:

(5x / 3(x-4)) * (x-4)(x-2)/(x(x-2)) which finally makes:

5/3

*****Remember that sometimes you cannot simplify an expression*****

****For example:****

***(x+4)(x-1)/(x-2)***

**Homework:**

*page 558 #4-48 even*

f(x)= p(x)/q(x)

p&q are polynomials

The easiest of all rational functions is:

y = x/1   line y = x  or  y = 1/x  x = o

y = 1/x

This graph shows y = 1/x

Horizontal Asymptote(HA): y = 0

Vertical Asymptote(VA): x = 0

X-Int.: None

Y-Int.: None

“We just graphed the easiest graph for a rational function” (Higgins) 4/8/2009

4 Identifiable Parts

x-intercept: when y = 0

y-intercept: when x = 0

vertical asymptote(VA):  what x cannot equal

horizontal asymptote(HA): what y cannot equal

Specail Rules for HA

If the power is the same on TOP & BOTTOM  y= (ax + b)/(cx + d) –> y = a/c

ex. 1. y = (5x^2 + 5)/(3x^2 + 3) –> y = 5/3

If the power is LARGER on BOTTOM  –> y = 0

ex. 2. y = (3x + 1)/(5x^2 + 3) –> y = 0

If the power is LARGER on TOP –> NONE   (think of Dolly Parton) 

Here are some exmples before graphing…find HA,VA, x-int, y-int

1. y = (x + 3)/(x – 2) –> y= 1/1

 or 1

HA: y = 1     VA: x = 2   

x-int: (plug 0 in for y)  0 = (x + 3)/(x – 2) –>                         (x – 2)*0 = (x + 3)/(x – 2)*(x – 2) –> 0 = x + 3 –> -3 = x

y-int: (plug 0 in for x)  y = (0 + 3)/(0 – 2)  –> 3/-2 = -1.5 

2. y = (-22)/(x + 43)

HA: y = 0   VA: x = -43

X-int: 0 = (-22)/(x + 43) –>                                                         (x + 43)*0 = (-22)/(x +43)*(x + 43) –> 0 = -22 <–None

Y-int: y = (-22)/(0 + 43) –> -22/43

3. y = (x + 3)/(2)

HA: None   VA: None

X-int: 0 = (x+ 3)/(2) –> (2)*0 = (x + 3)/(2)*(2) –>            0 = x + 3 –> x = -3

Y-int: y = (0 + 3)/(2) –> 3/2

Fun Fun Homework!!!

pg. 543 4-9 all 4 parts

11-19 HA & VA

59-69 odd

the main point is that if you have good genes/dna and have a healthier diet life will be better than the opposite….

when x = 1,2,3,4,5 and

y = 5,10,15,20,25

this is Direct Variation…when one goes up the other goes up.

when one goes down the other goes down.

when x = 1,2,3,4,5

and y = 5, 5/2,5/3,/5/4,1

this is Inverse Variation…When one goes up, the other goes down.

x = 1,1,2,-2,0

y= 1,2,1,-1,zebra

z=5,10,10,10,0

this is called joint variation…. Z depends on X & Y

Three Rules of Thumb

y = kx  “y varies directly as x”

y = k/x “y varies inversely as x”

z=kxy “z varies jointly as x & y”

“k” being a constant.

we did some examples….

xy = 1/7          xy\x = 1/7/x           <— revert and multiply   y= 1\7x             this example was inverse.

x = 5y      5y\5 = x\5 or y = 1\5x   this one is direct

x + y = -6.5   …….. this is none.

the homework was pg. 537, #’s         4-15, 21-28, 35-38, 40,41

Now that all that is out of the way, we can get to the real lesson.

ADDING

When you add in base ten you carry numbers over when they exceed nine such as

¹15

+17

32

Because the 7 and 5 add to something more than nine you carry over the 1 to the next column to add. You do the same thing in any base, only you carry when the number gets past one less than the base.

¹75₈

+64₈

161₈

Because the 5 and 4 add up to more than 7, you have to carry. They equal 9. Carry 1 over to the next column because 8 goes into 9 one time with one 1 left over in the first column. The same rule is used when the next column adds up to 14. 8 goes into 14 once with 6 leftover, since there is not a next column, just put the 1 in front as you would with base 10.

SUBTRACTING

When you subtract in base ten, if the number being subtracted in a column is more than the number it is being subtracted from, you borrow from the next number.

⁴5 2¹²

-27

25

Since the 7 is more than the 2 it is being subtracted from, we have to borrow from the 5 in front of it, adding 10 to the 2 making 12-7=5. The rest of the problem is self-explanitory.

²3 3₆ ⁹

-2 5₆

4₆

Because the 5 is more than the 3 it is being subtracted from, you have to borrow. When in base ten you borrow ten from the number in front of it because every place increases by ten. When in another base when you borrow, you borrow the number of the base.

MULTILPLICATION

Multiplying is the same as in base ten and set up the same way, only you have to follow the rules we learned in addition and carry when the number exceeds one less than the base.

¹3³14₆

*51₆

314₆

+24220₆

24534₆

When the numbers multiply to more than one less than the base, find out how many times the base will go into that number, carry that and whatever is left over will be what you drop down. If you multiply by a 2-digit number remember to put your 0 in!

DIVISION

To divide numbers in a base other than 10, you can do long division, or just change their bases to ten, divide them, then convert back to the original base.

654₇/12₇

When we convert them we get 333/9=37

Then we convert 37 back into base 7, which equals 52₇

HOMEWORK

Finish the worksheet from yesterday

“our” numbers are all in base 10

0   1  2  3  4  5  6  7  8  9  10

^^^ 10 digits

Then Higgy showed us examples of how number symbols evolved over time.

What about other bases like 5, 6, 7…?

0  1  2  3  4  <— 5 total digits (base 5)

0  1  2  3  4  5 <— 6 total digits (base 6)

0  1  2  3  4  5  6 <— 7 total digits (base 7)

Base 10 is our standard numbering system

Base 10 is expressed as:

There are 5 hundreds, 6 tens, and 2 ones.

Counting:

Base 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Base 8: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11… 17, 20,21… 26, 27, 30…

“Buckets of Fun”

Using these “buckets” can be very helpful when using bases

How to convert a different base to base 10:

How to convert base 10 to a different base:

More examples:

Homework:

Worksheet! (Numbers 1, 2, and the first #3)

2.  This is my favorite tie because it looks really neat and has a very spiffy color. Besides, anything that involves pi gets my attention. Pi is one letter away from pie, and let’s be honest, pie is delicious.

3 & 4. This tie involves pi. (That unintentionally rhymed)   Pi is the ratio of a circle’s circumference to it’s diameter. Pi is a number that continues infinitely and is irrational.There is no known repeating numbers in Pi. Many have tried to find these repeating patterns, but none have been discovered. In fact, it has been speculated that it is not even possible to have such repeating patterns. Pi is used in many mathematical equations, mostly those that involve circles or spheres.  

Pi facts:

Another fraction used as an approximation to Pi is (355 / 113) which is accurate to 0.00000849%

Pi is a transcendental number

Taking the first 6,000,000,000 decimal places of Pi, this is the distribution:

0 occurs 599,963,005 times,

1 occurs 600,033,260 times,

2 occurs 599,999,169 times,

3 occurs 600,000,243 times,

4 occurs 599,957,439 times,

5 occurs 600,017,176 times,

6 occurs 600,016,588 times,

7 occurs 600,009,044 times,

8 occurs 599,987,038 times,

9 occurs 600,017,038 times.

5. The ties I think Mr. Higgins should have:

^ This crazy math tie.

^ This binary tie (from ThinkGeek.com, and sadly to admit, I’ve actually bought things from there before…)

I chose this tie!

This one is a favorite of mine; mostly because I love the Indians, but baseball is also one of the best sports EVER! (next to softball of course)

A few facts about the Cleveland Indians:

• Cleveland was a charted member of the American League, winning its first home game 4-3 over Milwaukee on April 29, 1901.  They were originally dubbed the Blues, a name once used by Cleveland’s National League entry in the 1880s.
• The Indians contested for only one pennant (unsuccessfully) in the Lajoie era, but the second baseman was so popular, the team changed its name to the Naps in his honor.  After he departed in 1914, a newspaper contest led to fans renaming the team “Indians” in honor of the “Miracle” run that Boston’s Braves made to that year’s World Championship.
• The Tribe made it to the top in 1920, even with the stunning on-field death of their spark plug shortstop Ray Chapman.  He took a direct hit in the head on a fastball launched by New York’s Carl Mays on August 16.  Chapman collapsed at home plate, never regained consciousness and died the following morning.

There is some relevance between this tie and math.  The baseball is a sphere and the formula for the volume of a sphere is: V=4/3∏r^3

The books on the shelf are rectangular prisms and the volume for one of those is: V=lwh

The handle of the bat is a cylinder.  A cylinder’s area is: V=∏r^2h

And the flag in the background is a triangle and the area of a triangle is: A=1/2bh

I think this tie would be perfect for you to have!

Yay pink! Haha

The form of an exponential equation looks something like this:
y=b^x (where b is called the base)

To decide whether an equation has exponential growth or decay, you have to know the difference between the two.

exponential decay- occurs when quantity decreases at a rate proportional to its value

exponential growth- occurs when the growth rate of a mathematical function is proportional to the function’s current value

Going back to the equation y=b^x, by plugging in the base (b) you can determine decay or growth.

If b>1, it is exponential growth.
If 0<b<1, it is exponential decay.

The graph above represents exponential decay.
The rate at the x-axis is time, while the rate for the y-axis can be a number of things, but in this case it is the count rate.  Notice that the rate is decreasing, hence exponential decay.

The graph above represents exponential growth.
Again, the x-axis is time but the y-axis in this graph is bank balance.
The rate is increasing, hence exponential growth.

*the x-axis is always time

Parte Dois (Part Two)

Okay, logarithms can seem kind of confusing at first.  But if you look at it step by step, it’s pretty easy to understand.  Let’s start by answering your first question.

Q1: What are logarithms?

A1: In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.

Or in other words…

It’s an equation used to solve for a missing exponent.

Q2: Why do logarithms exist?

A2: Once you know what logarithms are, it makes more sense why they exist.  They exist so that you can solve for the exponent.

Q3: What does this mean?: log28=3  (said: log of 8 base 2 equals 3)
(The right half of the icons weren’t working.  And I promise it’s not me, it’s just my computer.) 

A3: Take your equation as an example:
log28=3

Another way of writing that is:
2^3=8

Since the log equation equals 3, you know that the 3 is the exponent.  This is because the point of logarithms is to solve for exponents.  Then you take the base of the log (in this case it’s 2) to what is on the right of the equal sign (3):  2^3

You know 2^3=8, and you could also figure that out because of the 8 in the original equation.

It usually makes more sense when you solve the equations instead of reading how to do it, so let’s do another example:

log416=2  (said: log of 16 base 4 equals 2)

2 is the exponent because it was what was solved for.

4 is the number in front of the exponent because it is the base of the log.

With this information, we can set up the equation:
4^2

And we know 4^2=16, which matches up with the log equation.
log416=2
-or-
4^2=16

Q4: How come I keep seeing problems like ln(5) that don’t use the word “log”?

A4: The equation ln(5) doesn’t used the word “log” because it is a “natural log”.  The difference between log and ln is that log has a base of 10 (when a base is not given) and ln has a base of e.  The set-up for ln equations and log equations are still the same.

I hope this helps!

Del Tre (Part Three)

These are a few websites I found that show a little history of mathematics:

http://aleph0.clarku.edu/~djoyce/mathhist/

http://www.maths.tcd.ie/pub/HistMath/  *

http://www-groups.dcs.st-and.ac.uk/~history/Indexes/HistoryTopics.html  *

http://en.wikipedia.org/wiki/History_of_mathematics  *

* website has other links for more information

Parte Quattro (Part Four)

Here is my “unique and interesting looking graph”.

My equations I used:

black:  y(x)= -3tan(x) red:  y(x)= -3cos(x)

blue:  y(x)= -5tan(x) green:  y(x)= 1cos(x)

orange:  y(x)= 3cos(x)

THE END!

Exponential growth or decay is created by plugging in values for the letters in the equation y = ab^x.

x is the rate of change and b is the base.

Exponential growth is when a graph (from an equation) grows upwards. This is created when a > 0 and the b is greater than 1.

Exponential decay is when a graph (from an equation) goes downwards. This is created when a > 0 and the b is between 0 and 1.

PART 2

Logarithms are just practically a different way of writing the equation: bⁿ = x. In logarithm form, this equation would be log_bx = n. Logarithms exist to solve equations like 2^x = 8. To solve this, you would write it as log₂ 8 = x. This is a simple example, but logarithms especially help for ones that are more complicated. So the statement of log₂ 8 = 3 means 2³ = 8.  (2×2x2 = 8). Ln is practically the same thing as log. Ln means natural log. Log has a normal base of ten (when no other number is in the base) and ln has a base of e. Some scientist argue that only ln exists, but either could be used in a statement.

PART 3

http://en.wikipedia.org/wiki/History_of_mathematics

http://www.math.tamu.edu/~dallen/history/m629_97a.html

http://www-history.mcs.st-and.ac.uk/history/Indexes/HistoryTopics.html

All three of these sites provide much information about different types of math and math aspects.

PART 4

Values are:

Red: cos (5x+353)

Black: Sin (5x)

Blue: tan (4x+445555342355554)

Orange: 4sin (34958724937543957439572948758384757583848589474384390743094307809459080934065928509832423447598347x)

Green: tan(3x+38749287592845725)

I chose this tie because I’ve seen this painting often but never bothered to look up anything about it.

It’s based on a painting called “The Persistence of Memory” by Salvador Dali (1931).

The idea of melting clocks supposedly came to Dali in a dream. This was about the same time scientists were developing the notion of Entropy as the “arrow of time”. Melting Clocks symbolize the eventualy heat death of the universe, where “time” as we know it will melt to a crawl.

4. I guess clocks represent a good example of modular arithmatic?

8:00+6 hours = 2:00 because 14 ≡ 2  mod  12

5. You have to pick up a 8-bit tie! (only availible as clip-on though)

This is the best Tie because tennis is my favorite sport, and this tie reminds me of going to tennis practice with my friends.

The tie has numerous tennis balls bouncing around on a brown and white court. Three out-of-date rackets are also laid on the court. (they are not boyerlicious rackets)The background is all black.

Tennis originated from a 12th century French game called paume (meaning palm); it was a court game where the ball was struck with the hand. Paume evolved into jeu de paume and rackets were used. The game spread and evolved in Europe. In 1873, Major Walter Wingfield invented a game called Sphairistikè (Greek for “playing ball) from which modern outdoor tennis evolved.

The U.S. Open history started its roots from when it was just an exclusive sport for the “high society”, in which the U.S. chamion would win $17 Million. Annually on average, over 600 men and women would come to ocmpete in this high cash event.

Yellow tennis balls were used at Wimbledon for the first time in 1986.

The first woman to win the US Open was Virginia Wade.  She defeated Billie Jean King 6-4, 6-4

The mathematical aspects of the tie are:

the the dimensions of the tennis balls, the parallel lines and perpendicular lines on the tennis court, the intersecting lines and angles in the strings on the rackets and the geometric shapes in the court (squares and rectangles)

Here are some ties that would be fun to wear:

haha:-D

Any quantity that grows or decays by a fixed percent at regular intervals is an example of exponential growth or exponential decay

Many real world phenomena can be modeled by functions that describe how things grow or decay as time passes.

There are two functions that can be  used to show the concepts of growth or decay in applied situations.  When a quantity grows by a fixed percent at regular intervals, the pattern can be represented by the functions: 

Growth: y=a(1+r)^x

Decay: y=a(1-r)^x

^{a= initial amount before measuring growth/decay}

^{r=growth/decay rate (%)}

^{x= #of time intervals that have passed}

 example of  Exponential Growth:

example of Exponential Decay:

 Part 2

the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number

log_bx = n   means   b^{n = x.}

During the 1500s or 1600s,  calculators didn’t exist.  To do multiplications, divisions, and root extraction with numbers having five or more digits required a lot of time and work.  Logarithms reduced the needed amount of work. Today Logarithms are very closely related to exponential functions.

Natural logarithms have the number called “e” as its base.  (e is named after the 18th century Swiss mathematician, Leonhard Euler.)  

e is the base used in calculus.  It is called the “natural” base because of certain technical considerations.

 log ₂ 8 = 3

^this is an example of a simple log form of a Logarithm

the exponential form of this equation is:

2^x=8

the value of the log is 3 because…. 2³ = 8

There are 2 special bases for Logarithms:

1.) log:    y=log₁₀X      =log X

2.) ln:     y=ln_eX          =lnX <this is Natural law of X

Both bases are solved the same way except the natural law of X is always ln_e  and the base for a normal logarithm is always log_10.

ln(5) is a simple logarithm example involving the natural law of X.

you can solve this example easily by typing in the equation on your calc. using the LN button.

Part 3

History of mathematics websites:

1.)http://www.gap-system.org/~history/Indexes/HistoryTopics.html

^shows history of mathematics from different countries! 

2.)http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html

^shows famous mathematicians from 1700 B.C. until 1940!

3.)http://aleph0.clarku.edu/~djoyce/mathhist/

^check out the 23 major mathematical problems page too!

Part 4

Equations of graph: 

black: y= -3cos(x+1)

red: y= -3cos(x+2)

blue: y= 3sin(x+2)

green: y= 3sin(x+1)

orange: y= -3cos(x)

1.

2.  This tie is really cool because it shows all the planets.. even though Pluto technically isn’t a planet anymore.

3. Planets were often used in ancient times to predict everything from wars to how to treat certain diseases. This tie shows the sun and all the planets in the Milky Way.

4. The planets are traveling in ellipses. This term means that instead of traveling in a straight line they move in curved paths, similar to an oval. Using math you can calculate how far each planet is from the Sun and also how far away they are from Earth. Many things in space helped to prove Mathematicians theories. For example in 1919 a solar eclipse helped prove Albert Einstein’s theory of gravity correctly predicted the bending of starlight. Space has also given us the speed of light. This allows you to make a calculation on how things will move compared to the speed of light.

5. These ties would be sweet:

]]> http://www2.norwalk-city.k12.oh.us/wordpressmu/halgebra2/2009/04/18/tie-project-19/feed/ 0 http://www2.norwalk-city.k12.oh.us/wordpressmu/halgebra2/2009/04/18/tie-project-18/ http://www2.norwalk-city.k12.oh.us/wordpressmu/halgebra2/2009/04/18/tie-project-18/#comments Sat, 18 Apr 2009 15:46:01 +0000 1alvarev http://www2.norwalk-city.k12.oh.us/wordpressmu/halgebra2/?p=1531

1. I like this tie because it has Bugs Bunny and Marvin the Martin playing baseball.

2.Bugs Bunny is 69 years old and was created in 1940. He debuted in the short film  A Wild Hare. Bugs Bunny’s catch line was “What’s up, Doc?” He had many nemeses such as Daffy Duck, Elmer Fudd, and many others. Bugs Bunny manly appeared in Loony tunes. He later became the mascot of Warner Bros.

3.Some mathmatical aspects of this tie is the line of the baseball field. The oppositelines are parallel to each other. There are also different shapes in this tie such as circles, and rectangles.

2. I am using this tie for my post because I like Looney Toons and Marvin the Martian, and because there are some hidden shapes in the tie.

3. The character on this tie is Marvin the Martian, who is a character from Looney Toons. He is often the antagonist in the shows he is in, involved in plots to blow up Earth and such, but he is a favorite of many because of his stupidity and ignorance in his attempts to destroy Earth.

4. The tie contains some hidden circles, shown by the red circle, and repetition, shown by the green circle.

Exponential Decay!!!

Part 1!

For both exponential growth and decay, you would use the formula y=b^x.  

In exponential growth b is greater than 1. The graph would grow. In the formula the exponent would be positive.

In exponential decay b is greater than 0 but less than 1. In a graph, it would decrease. In the formula, the exponent would be negative.                                     

Exponential Growth!!!