Honors Algebra II 0809

The 4 Basic Conics:

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)

You are now going to be asked to slove rational equations.

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

Begin with these:

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}

In the past, we have seen tationals such as:

(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*

A rational equation is defined as:

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

Our opening slides had some pictures in which we were instructed to figure out what the relationship was between the images.  These aren’t the exact pictures….

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

The first slide of today’s notes showed several random pictures of computers, a light switch, colored pencils, clocks, and donuts. All of these items are used or counted in a base other than ten. Computers and electronics use binary, colors use hexadecimal, base 16, clocks use base 6, 12, or 60 depending on how you look at it, and bakery items such as donuts are counted in base 12, or a dozen.

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

THE Standard Base

“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)

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

Part 1

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. 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.

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!!!