Time to say Goodbye

It seems like yesterday we have started our journey.
It was my pleasure having you as my students.
I wish you all the best during tomorrows exam and in your future endeavors.
Take care.
Mr.P

OH NEVERMIND!

I was looking at the Semester 2, Hahaha

To Mr. P

Dear Mr. P,

The links don't work :S

Conics: Parabolas on a Coordinate Plane

On Tuesday, we learned how to make parabolas on a coordinate plane.

Recall what you learned about parabolas last year. Remember the standard rules of a parabolas shape where if |a| = 1, then the shape would be normal but if |a| > 1 then the parabola would get narrower and if |a| < 1 then the parabola would get wider.

Recall the standard form of parabolas.

(y-k) = a(x-h)^2 or
(x-h) = a(y-k)

For our Conics unit, we have learned the General Form for a Parabola:

Ax^2 + Cx + Dy + E = 0 or
By^2 + Cx + Dy + E = 0

We use completion of squares to convert to standard form to find the vertex.

Example 1: Find the vertex, direction and shape of parabola, axis of symmetry, domain and range for each conic.

2x^2 + 4x - 4y +22 = 0

We can't find any data with this kind of equation so ee need to complete the square in order to solve this equation.

2x^2 + 4x - 4y + 22 = 0
2(x^2 + 2x) - 4y = -22
2(x^2 + 2x + 1) - 4y = -22 + 2
(2(x + 1)^2 - 4y = -20) / 2
(x + 1)^2 = 2y = -10

Rearrange the equation...

-2y = -(x+1)^2 - 10

Now isolate the y...

(-2y = -(x+1)^2 - 10) / -2
y = (1/2)(x + 1)^2 + 5

Now we can fill in the missing data with this equation.

Vertex: (-1, 5)
Direction: Upwards
Shape: Wide
Axis of Symmetry: x = -1
Domain: (-∞, ∞)
Range: (-5, ∞)

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

First we need to solve for m. We do this by plugging in the values of x and y.

(5)^2 + (-3) - 6(5) - m = 0
25 - 3 - 30 = m
m = -8

Now we can solve the equation.

y^2 + x - 6y - (-8) = 0
y^2 - 6y + x + 8 = 0
(y^2 - 6y + 9) +x + 8 - 9 = 0
(y - 3)^2 + x - 1 = 0
(x-1) = -(y-3)^2

Therefore, the vertex is at (1,3).

QED

Also a reminder that all assignments are due in Wednesday. Please submit any work that you have by then!

Good luck! :D

Conics - Circles on a coordinate plane

Hey everyone

Today we started our conics unit, and the first topic is circles on a coordinate plane. There are a few important equations to remember:

Standard form of a circle: x2 + y2 = r2 AND (x - h)2 + (y - k)2 = r2

General form of a circle: Ax2 + By2 + Cx + Dx + E = 0

Area of a circle: π r2

Circumference of a circle: 2 π r

Distance between two points: D = √(x2 - x1)2 + (y2 – y1)2

Midpoint between two points: MP = (x2 + x1) , (y2 + y1)
2 2


The most basic operation with a circle is to find the center and radius.
Ex.:
(x-3)2 + (y+4)2 = 48

we find the center by using the h and k values, and reading them as opposites.

Center @ (3, -4)

the radius is the square root of the number after the equal sign.

radius is √48


The second operation we worked on is finding the center, radius, domain, and range of a circle in general form.
Ex.
x2 + y2 – 6x – 10y – 2 = 0

We reorganize the equation so that all like terms are together.

x2 - 6x + y2 – 10y = 2

Then we complete the squares by dividing the C and D values by 2 and squaring them. We add these numbers to both sides.

x2 - 6x + 9 + y2 – 10y + 25 = 2 + 9 + 25

Then we solve each set for one binomial that is squared. Now it is solved.

(x - 3)2 + (y - 5)2 = 36

Next, we find the canter and radius the same way as before.

Center @ (3, 5)
Radius = 6

After sketching the circle, the points are at (-3, 5), (3, -1), (3, 11), and (9, 5). This helps us find the domain and range.

Domain = [-3, 9]
Range = [-1, 11]


The next operation is very similar, but has coefficients.
Ex.
16x2 + 16y2 – 8x – 143 = 0

16x2 – 8x + 16y2 = 143

16(x2 – 1/2x) + 16y2 = 143

We do the same thing with dividing the second number by two, and squaring it.

16(x2 – 1/2x + 1/16) + 16y2 = 143

Then we divide all the numbers in the equation by 16.

16(x2 – 1/2x + 1/16) + 16y2 = 144
16

We then continue to solve as normal.

(x – 1/4)2 + y2 = 9

center @ (¼, 0)
radius = 3


The last operation is using the distance and midpoint formulas.
Ex.
End points at (-2, 8) and (3, 4).

D = √(x2 - x1)2 + (y2 – y1)2
D = √(-2-3)2 + (8-4) 2
D = √(-5)2 + (4) 2
D = √25 + 16
D = √41

Radius = √41/2

MP = (x2 + x1) , (y2 + y1)
2 2

MP = (3 + 2) , (4 + 8)
2 2

MP = (½, 12/2)

MP = (½, 6)

Center @ (½, 6)

Then we plug it in to the formula:

(x – 1/2)2 + (y - 6)2 = √41/2

Happy Name Day Mr. Piatek !!

I bid you a joyful name day and wish you a prosperous future with many more name days to follow!

November Hall of Famer

And the winners are:

1 - Vince 5 points - 3 bonus points
2 - Kendra, Sharlyne, and Mark - 3 points - 2 bonus points each

PreCalculus
40S Probability of Independent and Dependent Events II
December 14, 2011
　
Example 1: If the probability of not winning a game is 21/37, find the probability of wining the game.

P(Winning)=1-21/37=16/37
　
　
　
Example 2: A Kickoff consists of team A and B taking turns kicking 50 yard field goals. The first team to kick a field goal wins. Team A has a probability of 2/5 of scoring on any one shot and team B has a probability of 3/5 of scoring on any one shot.

a) If team B shoots first, what is the probability of team A winning on its first shot?

Team A

P(Ascore)=2/5

P(Amiss)=3/5 　

Team B

P(Bscore)=3/5

P(Bmiss)=2/5

P(Awin)= 2/5 * 2/5=4/25
　
　
　
b) If team A shoots first, what is the probability of team B winning on its second shot.
3/5*2/5*3/5*3/5
=54/625

Example 3: What is the probability of getting a sum of 3 on single throw of two dice.
Sample Space For Throwing Two Dice=36
1+2=3
2+1=3
　
P(Sum Of 3)=2/36=1/18
　
　
Example 4: What is the probability of getting a jack or a diamond on a single cut of a deck of 52 cards.

P(Jack)= 4/52

P(Diamond)= 13/52　　　　

P(JackDiamond)= 1/52

P(Jack or Diamond )= (4/52+13/52)-1/52

=17/52-1/52=16/52
　
　
　
　
　
　
　
　
　
　
　
　
　

Hey guys, this is Irish and today we learned about the “Probability of Independent and dependent events”.

Mutually Exclusive events are where events can not both occur at the same time. (The probability of drawing a king or a queen when a single card is drawn from a deck of cards – the card can not be king and queen at the same time).

• Probability for mutually exclusive events: P(E or F) = P(E) + P(F)
• Probability for non-mutually exclusive events: P(E or F) = P(E) + P(F) – P(E and F)
  

Complementary events are when one event occurs if and only if the other doesn’t.

• Complementary events: P(E) + P(F) = 1
  

Independent Events are when two events occur but do not influence the outcome of each other.

• Probability of Independent Events: P(E and F) = P(E) • P(F)
  

NOTE: DON’T REDUCE UNLESS OTHERWISE STATED

Example #1: Two cards are drawn from a desk of 52 cards. Find the probability that they are both face cards if the first card is…

a) replaced?
PFACE CARD (#1) = 12/52
PFACE CARD (#2) = 12/52
PBOTH FACE CARD = 12/52 • 12/52 → 9/169


b) not replaced?
PFACE CARD (#1) = 12/52
PFACE CARD (#2) = 11/51
PBOTH FACE CARD = 12/52 • 11/51 → 11/221

Example #2: A box contains 5 chocolate candies, 3 cherry candies and 7 caramel candies…

a) if one candy is chosen, what is the probability it will be cherry or caramel?
PCHERRY = 3/15
PCARAMEL = 7/15
PCHERRY and CARAMEL = 3/15 + 7/15 → 10/15

b) if two candies are drawn what is the probability that both are chocolate or both cherry? (without replacement).

PCHOCOLATE (#1) = 5/15
PCHOCOLATE (#2) = 4/14
PBOTH CHOCOLATE = 5/15 • 4/14 → 20/210

PCHERRY (#1) = 3/15
PCHERRY (#2) = 2/14
PBOTH CHERRY = 3/15 • 2/14 → 6/210
PCHOCOLATE and CHERRY = 20/210 + 6/210 → 26/210

Sample Space

Hey everyone, so here's our lesson from 12/6/11.

So, what is a Sample Space.

Well it is the complete set of all possible outcomes and is also useful in quickly determining the number of a particular outcome in an experiment.
To represent a sample space, tree diagrams and ordered pairs are often used.

Basic Probability Formula: Used to identify the probability that specific events will occur
P(E)= Success .
Sample Space

Example 1: A coin is flipped 3 times, sketch the sample space using a tree diagram.

Example 2: Using example #1, find the probability of tossing two heads.

P(E)= Success .
Sample Space

P(2 Heads) = 3 .

8

Hall of Famer of November

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

Permutations - part 1

Hey everybody, this is Irene, and our lesson on November 22nd was about Permutations.

The definition of "PERMUTATION" is a set of distinct objects in an arrangement of objects without any repetition in a specific order.

And to "PERMUTE" a set if objects means to rearrange them.

The formula we use for Permutation is:

nPr = n!

________

(n-r)!

Where:

n = total number of objects

r = number objects selected

Hint: n > r

( n; must always be positive.)

Ex #1: How many three letter "words" composed from the 26 letters in the alphabet are possible if . . .

a) NO repetitions are allowed?

Method 1; (can only use formula is NO repetitions are allowed)

n = 26; because you have 26 letters to choose from.
r = 3; because you're trying to make 3 letter words.

sol'n

Method 2: "dash method" (can apply in both no repetition and repetition situations)




B) Repetitions allowed:

SPECIAL NOTE:

(Understand)
Homework: Permutations, Combinations & Binomial Theorem Assignment

due TOMORROW!

Factoral Notation

COUNTING PRINCIPLES:

This unit is all about counting… without actually counting.

FUNDAMENTAL COUNTING PRINCIPLE

If something can be chosen, or can happen, or be done, in m different ways, and, after that has happened, something else can be chosen in n different ways, then the number of ways of choosing both of them is m · n.

MULTIPLICATION PRINCIPLE:

ü When two events are dependent we multiply their results to find the total number of ways it occurs.

Key Words: “AND”, “BOTH”

ADDITIVE PRINCIPLE:

ü When two events are independent we add their results to find the total number of ways it occurs.

Key Words: “OR”, “EITHER”, “OPPOSITE”

EXAMPLE:

The executive of the Manitoba Association of Mathematics Teachers Association consists of three woman and two men. In how many ways can a president and a secretary can be chosen if:

1. The president is to be a female and the secretary male?

Using Dash Method:

3 and 2 6

President Secretary

2. The president is to be a male and the secretary female?

2 and 3 6

President Secretary

3. The president and secretary are to be of the opposite sex?

6 and 6 12

President Secretary

FACTORIAL NOTATION:

The symbol n! ( read as n factorial) means to multiply all of the positive integers from n consecutively down to 1

The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples:

• 4! = 4 × 3 × 2 × 1 = 24
• 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040
• 1! = 1

*n! = n(n-1)(n-2)…(3)(2)(1) , where n is an element of + integer.

EXAMPLE:

ü 24!(n+1)!

22!(n-1)!

ü 24 •23•22!(n+1)(n+0)(n-1)!

22!(n-1)!

ü 552 (n+1)(n+0)

1

ü 552(n+1)n