last post for 2011!!!

today we knew how to find the equation for a parabola

EX. find the equation for the horizontal parabola with vertex at 4,1 and passing through the point 2,5. you might want to label your given points h,k and x,y for you not to get confused.

use the formula (x-h)=a(y-k)^2 from here all you have to do is plug in the values and solve.


we also knew how to find the vertex of a parabola if we are given an equation and a passing point.

EX. if the parabola y^2+x-6y-m=0 passes through the point (-3,5) find the vertex.

first step is to label the given passing point x and y for you not to get confused, next is to plug -3 to the x in the equation and 5 to the y's in the equation and solve for m. in this case your m shall equal to 8. next thing for you to do is plug your 8 for m to the equation. DO NOT PLUG -3 AND 5 TO YOUR x AND y, complete the square and solve algebraically. your vertex should be at (1,3). that's all and thank you.

Conics

In mathematics, a conic section (or just conic) is a curve obtained by intersecting a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2.

Circle




The equations for circle are :


The General formula for circle is = Ax^2 + By^2 + Cx + Dy + E = 0

Distance

Distance Formula is :



It is use to find the distance between the points and with that we can find the radius of a circle given the points of it.
Midpoint

Midpoint formula:



We can use this formula to find the center of the circle.
Parabolas



There are vertical and horizontal parabolas.

Vertical Parabola

We already know that the vertical parabola has an equation of :
y = a((x-h)^ 2 ) + k

where a is the coefficient that is responsible for the horizontal size of the parabola and whether the parabola opens up or down
where h is read as opposite and is vertical symmetry and a point in x axis
where k is read as is and a point in y axis

Example:




Horizontal Parabola.

Our normal parabola opens up or down, but this parabola opens to the left or right. The equation is:
x = a((y-k)^2) + h

Here's an simple equation and example of a Horizontal Parabola



Remember that in Vertical Parabola if a is positive the parabola goes up and down if negative and in Horizontal Parabola if a is positive it goes to the right and left if negative.

So that's mostly it.
Here's a circle, not an ordinary circle though. I don't know if you know this already. If you don't, look at the center of the circle and fixate your eyes at it then try looking at it closer then further, then closer, then further, over and over again. ;)



It moves right? :O lool



1 more day to goooo!! :D You know what it is ;P




4 more days 'til Christmas! Merry Christmas Everyone. ;)




-Paul

November Hall of Famer

And the winners are:

1 - Roxanne - 7 points - 3 bonus points
2 - Carjelu and Gurvinder - 6 points - 2 bonus points each
3 - Karla - 4 points - 1 bonus point

Sample Space

Hi fellow PreCal classmates! This is Paulene J Our blog site is not up to date anymore just because I forgot to post mine on time & I'm very sorry about that!

Anyways.. last December 06, we started a new lesson called Sample Space. It’s a very short topic under the Probability Chapter/Unit.. and here’s a recap! J

Sample Space refers to the complete set of all possible outcome. These are often represented using tree diagrams and ordered pairs. The probability that a specific event will occur can be described as.. P(E) = Success ÷ (sample - space)

There are also two events that can occur;

- Dependent events is when the outcome of one event affects the outcome of a second event.

- Independent events is when the outcome of one event does not influence the outcome of a second event.

November Hall of Famer

Please cast your votes (under comments) for the Hall of Famer of November.
Thank you.

Binomial Theorem

Hey fellow students ! last Friday we talk about binomial theorem which is discovered by a very smart man whose name is pascal . This theorem is basically a much quicker or less slower way of expanding a binomial expression that is raised to a very high power . To help us understand it i try to look for a site that can explain it better than me . So this what i found

http://www.math10.com/en/algebra/probabilities/binomial-theorem/binomial-theorem.html.

and i also found a very helpful video (: i hope this will help you guys even more .
http://www.youtube.com/watch?v=bMB8qDYa8N0

Combinations

The formula for the number of possible combinations of r objects from a set of n objects:



Example: How many different committees of 4 students can be chosen from a group of 15?

There are 1365 different committees.

	If the order doesn't matter, it is a Combination.
	If the order does matter it is a Permutation.

1.) Combinations with repetition

Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla. You can have three scoops. How many variations will there be?

Let's use letters for the flavors: {b, c, l, s, v}. Example selections would be

• {c, c, c} (3 scoops of chocolate)
• {b, l, v} (one each of banana, lemon and vanilla)
• {b, v, v} (one of banana, two of vanilla)

(And just to be clear: There are n=5 things to choose from, and you choose r=3 of them.
Order does not matter, and you can repeat!)

where n is the number of things to choose from, and you choose r of them
(Repetition allowed, order doesn't matter)

(5+3-1)!	=	7!	=	5040	= 35
3!(5-1)!		3!×4!		6×24

2.) Combinations without repetition

where n is the number of things to choose from, and you choose r of them
(No repetition, order doesn't matter)

By: Roxanne Ching

Circular Permutations

- are arrangements of elements in a circular pattern.

Formula:

Example 1:

• In how many ways can 5 people be seated around a circular table?

n=5

Formula: (n-1)!

(5-1)!

4!

4.3.2.1

= 24

• In how many ways can 8 girls be seated in a dinner table?

n=8

Formula: (n-1)!

(8-1)!

7!

= 5,040

* There are two cases of circular permutation;*

(a) If clockwise and anti clock-wise orders are different, then total number of circular-permutations is given by (n-1)!

(b) If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by (n-1)!/2!

Example 2:

How many necklace of 12 beads each can be made from 18 beads of different colours?

clock-wise and anti-clockwise arrangement s are same.

total number of circular–permutations:

¹⁸P₁₂/2x12

=18!/(6 x 24)

= 4.45x10 to the power of 13

* Some permutation problem have restrictions that affect the

answer and the process of solving it

• In how many ways can 8 people be seated around a circular table if Edgar and EJ refuse to sit next to one another?

STEP 1: Calculate total arrangements that is given in the problem.

n=8

Formula: (n-1)!

(8-1)!

7!

=5040

STEP 2: Calculate the number without following the restrictions, for this calculate the number when Edgar and Ej sit together.

*Edgar and Ej is going to be a one group because they sit together, so it makes (2!)

*added to the people left. (6!)

= 7!

(n-1)

(7-1)! =6!・2! = 1440

STEP 3: Calculate the number following the restrictions, for this calculate the number when Edgar and Ej is not sitting together.

* Subtract the total number of arrangement and the total number when Edgar and Ej is sitting together .*

5040-1440 = 3600

Permutations with Case Restrictions

Example 1:

*In how many 5 letter “words” are possible using the letters in BUFFALO?

*used the dash method

*count how many letters are their in the word BUFFALO. so theres 7 words.

*find letters that are repeated and divide the total number of with the number of letters that are repeated. and theres 2 repeated letters.

(7・6・5・4・3) ÷ 2 = 1,260

• Theres a set of basket of fruits being arranged in a cabinet. The first set has 3 oranges, the second has 4 bananas and the third set has 5 apples. In how many ways can the basket of fruits can be arranged if they should be kept in a same cabinet together?

○ - oranges

▲- apple

～ - bananas

～～～～ 4!

▲▲▲▲▲ 5!

○○○ 3!

*multiply all the number of quantity of the given number of fruits and the number of item that are given.

4! ・ 5! ・3! ・3! = 103,680

Hi Everyone!

1. Problem: What is the probability of rolling a 3 on a die (plural, dice).

Solution: On a fair die (not the kind you play with in Vegas, where everything is rigged), there are six equally likely outcomes when you roll. Also, there is only one way to get a 3. By the definition of probability, P(3) = (1/6).

A permutation of a set of objects is an arrangement of the objects in certain order. For example, take the set of four objects {pepperoni, sausage, onions, mushrooms}.

They can be arranged on a pizza many different ways. Below are a few of the
ways.

pepperoni, sausage, onions, mushrooms sausage,
onions, mushrooms, pepperoni onions, mushrooms, pepperoni,
sausage mushrooms, pepperoni, sausage, onions pepperoni,
sausage, mushrooms, onions There are some more, but we won't list
them.

To find the number of different arrangements of the set we select a first
choice; there are 4 possible choices.

Now we take a second choice; there are 3 choices. Now pick a third choice; there are 2 choices.

Finally, there is 1 choice for the last selection. Thus, there are 4
* 3 * 2 * 1 or 24 different ordered arrangements of the toppings.

This product can also be written as 4! (read: 4-factorial).
The total number of permutations of a set of n objects is given
by n!. Example:

1. Problem: 5!
Solution: 5 * 4 * 3 * 2 * 1
120
When you have a set
of objects and only want to arrange part of them, you have a permutation of
n objects r at a time.

For example, if you have 6 toppings for a pizza, and a customer calls and tells you to put any 3 toppings on the pizza, you might want to know how many different pizzas you can make. You can select the first topping in 6 ways, the second in 5, and the third in 4. As we learned above, this can be written as 6 * 5 * 4. There is a theorem that tells us about a formula for the situation above. It says the number of permutations of a set of n objects taken r at a time is given by
the following formula: nPr = (n!)/(n - r)!.

Example:

2. Problem: If a school has lockers with
50 numbers on each combination
lock, how many possible
combinations using three
numbers are there.
Solution: Recognize that n, or the number
of objects is 50 and that r, or
the number of objects taken at one time is
3. Plug those numbers in the permutation
formula.
50!
50P3 = --------
(50 - 3)!
Use a calculator to find the
final answer.
117600
Things are immensely simplified when you can repeat the objects. For example, if you are making license plates with only 4 letters on them, and you can repeat the letters, you can take the first letter from 26 options, the same for the second, third, and fourth. Therefore, there are 264 or 456976 available
license plates using 4 letters if you can repeat letters. There is a special
theorem that tells us the number of arrangements of n objects taken
r at a time, with repetition is given by nr.
Example:

3. Problem: How many 4 digit license plates
can you make using the numbers from
0 to 9 while allowing
repetitions.

Solution: Realize there are 10 objects
taken 4 at a time. Plug that
information into the formula for
repeated use.
104
10000

Applications of Exponential Functions

Today in class we learned about Applications of Exponential Functions.

There are three Formula's you should know

Formula for Growth and Decay

A=Pe^rt Where: A = Final amount, P = Initial amount, r = rate/constant, and t = time

Formula for Compound Interest

A=P(1+r/n)^rt Where: A - Final Amount, P = Inital Amount, r = Rate, n = number of compounds per year, and t = time.

Formula for Continuous Compound Interest

A=Pe^rt Where: A = Final amount, P = Initial amount, r = rate/constant, and t = time

Now that we know those questions lets do a couple questions.

Example 4 from the Exponents and Logarithms booklet.

A $15000 investment earns interest at the annual rate of 7.6% compounded quarterly.

Determine the value of the investment after 9 years

and the amount of interest earned after 9 years.

1. State what is given.

Initial amount = 15000

Rate = 7.6% = 0.076

time = 9 years

and n = 4

2. Plug in your values into your formula.

A= 15000(1+ 0.076/4)

A=15000(1.019)^36

The ^36 came from multiplying 9 by 4 since you are looking for the investment after 9 years and you are being compounded quarterly(4)

3.A= 29536.70

You have found out that the value of your investment is $29536.70 after 9 years. Now you want to find out the interest earned after 9 years.

4. Interest earned = Final Amount - Initial Amount

5. 29536.50 - 15000 = $14536.7

Example 2

Bacteria are growing according to y = A * 2^x. If the initial bacteria count is 10,000 determine how many hours it will take for the bacteria count to triple.

1. State what is given

A=10,000

final amount = 30000( since you know that the amount your looking for is tripled you mulitply your original initial amount which is 10,000)

time=x=?

The formula you would use would be A=Pe^rt

2. plug in your amounts

30000=10,000 * 2^x

divide both sides to get 2^x by it's self

3. 3=2^x

you can't make bases the same so you have to put logs in the front

4. log3 = log2^x

divide by log 2^x to get the x by it's self.

5. x= log3/log2

6. now you type in log3/log2 in you calculator and you should get 1.6 rounded off .

Thanks for reading and just a reminder that we have a test on Friday so start studying!

Logarithmic Theorems II

Hello,

        These Are the Logarithmic Simplification Steps
     1. Rearrange the terms- moving all negative to the end    
     2. Exponent Law- Coefficients become exponents (move to the top)
     3. Multiplication Law- Addition becomes multiplication
     4. Division Law - Subtraction becomes division
     5. Fractional Exponents become roots

     Change of Base Formula:

       Note:
            - Calculators are based on 10
            - If not, we assume base 10
            - if it isn't on base 10, we will need to change the base to 10 so we can evaluate logarithms using the 
             change of base formula

Example 1:

     
Example 2:

Example 3 (Evaluate to 4 Decimal Places)

That's all and Have a great night!!!!

Logarithmic Theorems 1

Laws of Logarithims

Division Law: loga M/N = loga M - loga N For Division Law switch the division into subtraction

Multiplication Law: loga MN = loga M + loga N For Mulitiplication Law switch the multiplication into

addition

N

Exponent Law: loga M = N loga M Just put the N to the Front

Logarithmic Expansion Steps

1. Roots become fractional exponents

2. Division Law - Division becomes subtraction

3. Multiplication Law - Multiplication becomes addition

4. Exponent Law - Exponents become coefficients (move to the front)

Expand

logb Division becomes subtraction

= logb - logb x Roots become fractional exponents

2 1/2

= logb (x - 1) - logb x Exponents become coefficients (move 1/2 to the front)

2

=1/2 logb (x - 1) - logb x This could be simplified further

= 1/2[logb (x+1) + logb (x-1)] - logb x Multiply the brackets

= 1/2 logb (x+1) + 1/2 logb (x-1) - logb x Final Answer

Here is another example:

If logb 2= 0.3010 , logb 3= 0.4771 , and logb 5 = 0.6990 , then:

logb 20/3 = Division becomes subtraction

= logb 20 - logb 3 logb 20 can be simplified

= logb (5)(4) - logb 3 4 can be simplified further

= logb (5)(2)(2)- logb3 Multiplication becomes addition

= logb 5 + logb 2 + logb 2 - logb 3 Switch with the other values

= 0.6990 + 0.3010 + 0.3010 - 0.4771 Solve

= 0.8239 Final Answer

I hope this helped. Brendan

October's Halll of Famer

Please cast your vote for the Hall of Famer of October.

Exponential functions

Hey guys what's up ! Mehtab here. I'm going to try to solve some exponential functions and i hope that it might help you as well. These functions needs to be moved vertically or horizontally whenever required. We have done the transformation unit, so this is not going to be a hard one, I guess. So,

Important :

.

For example:

Let's start by using table of values, the easy method.

After that you graph it, simple as that. There is no horizontal asymptote, so y=0.

Another example:

Table of values:

No horizontal asymptote, so y=o.

(B)

Now, graph and then move all the points to the right by 3, resulting in the end of your question. I hope you all got it.

Double Angle Trigonometric Identities

Hey everyone!

For today's lesson, we studied Double Angle Trigonometric Identities which is very simple and I know that a lot of you have no troubles with it. So I'll be very brief on explaining how this one works.

Basically, this lesson is very similar to our previous lessons. The process and steps are the same.

The only thing you have to do is to find the equivalent value of the given identity from your formula sheet and substitute it to the original function.

Here are the basic Double Angle Trig Identities:

You can have different forms of equations but it will end up with the same answer as long as you don't mess up. Always choose the easier ones.

On some questions, you'll be ask to find the exact values of sin, cos or tan. Always follow the simple steps. Once you get the equation, set alpha and beta. Solve and find the value of trig identity. You need to use CAST rule, Special triangles and Pythagorean theorem to find the values depending on what you are told to look for.

As long as you know the equivalent of each set of identities(which is in our formula sheet), you'll be fine.

Hope this helps!

K bye.

Homework

Hi,Does anybody know what is the homework for today? Please reply and Thank You.

-David

Sum and Difference Part 2

Hi Guys..

Since there is no new topic and we only did exercises in the booklet... I'm just gonna answer the last question it this topic...

Hope this helps...

Sum and Difference Identities Part One

HI!!! Today in precal we studied about sum and difference identities part one.
lets go to the first example

ex. sin 7pie/12 first step that you need to do is to set an equation that will equal to 7pie/12
the equation that you need to have is either addition or subtraction. note that there is many possible equation for the example given.

sin 7pie/12=sin(4pie/12+3pie/12) *SIMPLIFY*
=sin (pie/3+pie/4) *LABEL as ALPHA and BETA*
alpha beta
at this point you have to have your formula sheet with you. The formula for this equation would be sin(A+B)= sinAcosB + cosAsinB meaning you should have sin pie/3 x cos pie/4+cos pie/3 x sin pie/4 what you have to do next is to find the exact values of these trigonometric values which means you need to use the special triangles and the cast rule. therefore you should have...





your final answer should be as stated in the image above. don't do anything to your final answer leave it as it is.

at some point you will be given csc 19pie/12. the trick to this is to solve it as sin 19pie/12 and just get the reciprocal of your final answer.

K BYE!!! ahaha...

TRIGONOMETRIC IDENTITIES

HELLO!

We started on discussing about trigonometric identities. It is basically proving that both sides of an equation are equal. To help you do this, you should always consider the following identities:

sin θ	=	1 csc θ		csc θ	=	1 sin θ
cos θ	=	1 sec θ		sec θ	=	1 cos θ
tan θ	=	1 cot θ		cot θ	=	1 tan θ

a)	sin²θ + cos²θ	=	1
b)	1 + tan²θ	=	sec²θ
c)	1 + cot²θ	=	csc ²θ

tan θ =

sin θ cos θ

cot θ =

cos θ sin θ

For example,

Answer

We express everything in terms of sin y and cos y and then simplify:

Just play around with the equation until you end up having both sides equal. BUT STILL MAKE SURE THAT YOU ARE PLAYING IT RIGHT!

More examples

Happy Birthday Mr.P

Have a Happy Birhtday!

Hi guys! Last friday, we were finishing up the topic about Sinusoidal Functions.

Let's take a recap about how to create equations for sinusoidal functions.

1. First, we have to find the middle axis (this will be d-value)
   
2. Find the amplitude (this will be the a-value). The easiest way to find the amplitude is to count from the middle axis to the highest point the graph is reached)
   
3. Determine the period and then calculate the b-value.
   
4. Identify the type of original wave (it's either y=sinx or y=cosx)
   
5. Create the first equation using the a, b, c and d values
   
6. Create the second and third equations including the a,b,c and d values.

So for example.

1. The middle axis will be at 0 which makes the d value = 0

2. Amplitude = 1

3. Period = 2π so b value will equal to 1

4. The type of original wave for this graph would be y=cosx

5. So now, we will create our three equations

I. y=cosx

II y=sin(x-)

iii. y=-sin(x+)

Note: Your goal is to manipulate the type of function you`re working with for the second and third equations by making it mask the original equation. Also, when creating the two other equations after determining the equation for the sinusoidal function, the two equations have to be the opposite or the other function that was not the original. If the original function was cos, then the two other equations would be sin. And vice versa.

The other thing we learned about is solving functions which involves some of the stuff we learned from Unit 1: circular functions

For example: p(x)=-cos(x+5) - 2

Things to keep in mind: special triangles and quadrantals.

p(0) = - cos(0+5) - 2 I put x to 0

= - cos(5) - 2

Multiply 5 by which becomes ⁵

= -cos ⁵ - 2

** Cos = adj/hyp so ⁵ = 1/2 but since cos was -'ve then 1/2 would be -'ve as well.

= -1/2 - 2

= -1/2 - 4/2

= -5/2

Quick reminder: Every lesson we learned will be helpful to the next unit and so on.

Anyways, hope everyone had a greaaat weekend!! :)