Reflection

This week we continued working on the review test in class, and hopefully we almost done with them. In chapter 13 we learned about sequences, many formulas, sigma notation, etc.
Sigma Notation

100 - 1 & 100 are called the limits of summation.
sigma n^2 - n^2 is the summand
n=1 - the bottom variable is called the index, which is K.

- To evaluate-plug in the numbers between your limits of summation into the summand.
Adding each term to form a series.

- Examples:
Give each series in expanded form

1) 4
sigma 5K = 5+10+15+20
K=1

2) 6
sigma 9+16+25+36
n=3

Express the series in sigma notation

1) 1+2+4+8+16+32
5
sigma 1X(2)^K
K=0

2) 48+24+12+6+...
infiniti
sigma 48(1/2)^c
c=0

For some reason i am still having trouble with sequences even though it super easy, i guess i need to know the formulas better.

Reflection March 28

This week went by slow, even though we had a field trip wednesday for biology and got out early tuesday for softball. Then, we got our report cards on thursday, and i did pretty good. I would like to do better..I got all a's and b's and 2 c's. And one of those c's is in this class:/ I really need to get that up to a b so that i will be that much closer to getting all a's and b's. And then the spring fest is this weekend, wahoo, hah. All we have been doing in class is doing the chapter tests and stuff like that.....soo, heres some stuff from chapter 9....

A) a=2, b=4, angle A=22
sin22/2=2sinB/4
4sin22/2=2sinB/2
sinB=4sin22/2
B=sin-1 ((4sin22)/(2))
B=48.522

B) b=3, c=6, angle B=30
sin30/3=sinC/6
6sin30/3= 3sinc/3
c=sin1-((6sin30)/(3))

C) a=7, c=5, angle A=68
sin68/7=sinc/5
5sin68/7=7sinc/7
c=sin-1 ((5sin68)/(7))
c=41.474

Reflection 3/26

Okay, so here's another week that B-rob was not here and we did absolutely nothing in class. We made up some other chapter tests, but that is about it. We had a sub for the rest of the week and we were only in class four days, because biology II had a field trip. Most of the week was just sitting around doing any work that you could find because spring fest is this weekend too! I have no clue what we are going to learn when we come back on monday. I'm just excited that i got a B on my report card in Advanced Math! So i guess i'm just going to review some more because i don't know what we're going to learn next week.
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Here are some formulas that we learned in chapter 10:


Cos(α +/- β)=cos α cos β -/+ sin α sin β
sin(α +/- β)=sin α cos β -/+ cos α sin β
sin x + sin y= 2 sin x + y/2 cos x-y/2
sin x - sin y= 2 cos x + y/2 sin x-y/2
cos x + cos y= 2 cos x + y/2 cos x-y/2
cos x - cos y= 2 sin x + y/2 sin x-y/2

tan (α + β)=tan α + tan β/1-tan α tan β
tan (α - β)=tan α - tan β/1+tan α tan β

sin2α=2sin α cos α
cos 2α=cos^2 α –sin^2 α = 1-2 sin^2 α= 2 cos^2 α -1
tan 2α = 2tan α /1-tan^2 α
sin α/2= +/- √1-cos α/2
cos α/2= +/- √1+ cos α/2
tan α/2= +/- √1-cos α or 1 + cos α
=sin α/1+cos α
=1-cos α/sin α
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This is also some other Laws that we learned in that chapter:

LAW OF SINES

sinA/a = sinB/b = sinC/c

LAW OF COSINES

(opp leg)^2 = (adj leg)^2 + (other adj leg)^2 -2(adj leg)(adj leg)cos(angle between)



Example:

x= |6^2 + 5^2 -2(5)(6) cos 36*

x=3.530

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Here is the change of base formulas:

Changing Bases
....use when a log cannot be solved
....used to solve for x as a variable
....used to change the base of log

1. exponetial form
2. take log of both sides
3. move exponents to the front
4. solve for variable
5. write as a function or whole number

Example:

log base 3 of 7
1. 3^x=7
2. log 3^x = log 7
3. x log 3= log 7
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We also learned what the different quadrants mean, and how to switch from each of them:

QUADRANT RULES:

I to II---make it negative and add 180 degrees
I to III---add 180 degrees
I to IV---make it negative and add 360 degrees
II to IV---add 180 degrees


Example:

cos^-1 (-1/2)

Cosine is negative in the II and III quadrants
Your reference angle is given as 60 degrees.
I to II---make 60 negative and add 180 degrees = 120 degrees
I to III---add 180 degrees to 60 degrees = 240 degrees
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I guess no one gets to comment on my blog because i understood everything that i blogged about, but if anyone wants to put more examples of the stuff that i blogged about, they can. But other than that, spring fest is probably going to be gay. kbye:)

Comments

This week has flown by and state is finally here. I'm more pumped about it than i thought i would be. This should be pretty fun. Then spring fest and a whole week off. The next week and a half is going to be great.

Reflection 3/21

Triangle Formulas:

A= 1/2 (leg) (leg) Sin (angle between). This is used for non-right triangles.

For Area say you have a triangle with:

A side length of 4
A side length of 5
And an angle of 30 degrees

Then plug it all in: A=1/2 (4) (5) Sin 30 degrees

A= 10 Sin 30 degrees which aproximately = 5.


For another triangle:

A side length of 3
A side length of 8
And an angle of 60 degrees

Then plug it all in: A= 1/2 (3) (8) Sin 60 degrees

A= 12 Sin 60 degrees which aproximately = 10.392

This formula can only be done when you have two given lengths and at least one angle.

Reflection #31

Okay, I'm going to review law of sine and cosine:

LAW OF SINES:
sin A/a = sin B/b = sin C/c
**used when you have a pair

Example:
Visualize a triangle ABC:
A = 47 degrees
a = 8
B = 92 degrees
Find b.
sin 47/8 = sin 92/b
Cross multiply.
b sin 47 = 8 sin 92
Divide.
b = 8 sin 92 / sin 47
Plug into calculator.
b = 10.932

LAW OF COSINES:
(opp. side) = (adj. side)^2 + (other adj. side)^2 - 2(adj. side)(other adj. side) cos (angle between)
**used when you can't use law of sines

Examples:
Visualize a triangle ABC:
A = 78 degrees
b = 8
c = 3
Find a.
a^2 = 8^2 + 3^2 - 2(8)(3) cos 78
Square root both sides and plug into calculator.
a = 7.939

I guess B-Rob comes back Monday, but I won't be there...I'm sick unfortunately. So if anyone wants to comment about what we learn Monday, that'd be really helpful. ^^

Reflection.

Chapppterrrr 11!

*polar is in the form (r,theta) or r=__theta
*there are directional differences with r
*cos(theta)=x/r
*sin(theta)=y/r

**to convert from polar to rectangular>>> x=rcos(theta)
**to convert from rectangular to polar>>> y=rsin(theta)

***to convert from rectangular to polar>>> r= +- square root of x^2 + y^2
theta= tan^-1 (y/x)

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EXAMPLE:

Give the polar coordinates for (3,4).

r= +- square root 3^2 + 4^2
r= +- 5

theta= tan^-1(4/3)
theta= 53.130degrees

*which quadrant is (3,4) in? <<<< I
THEREFOR---> the angle in quadrant I goes with the positive 5.

(5,53.130degrees) and (-5,233.130degrees)

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To convert equations:

1. To go from rectangular to polar, Plug x=rcos(theta) and y=rsin(theta) into the equations, and solve for r.

2. To go from polar to rectangular, Use identites to get rid of the number in front of theta. Plug in y/r for sin(theta) and x/r for cos(theta). Plug in square root of x^2 + y^2 for r. Solve for y if possible.

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EXAMPLE:

x^2 + y^2 = 1 <---- convert to polar


(rcos(theta))^2 + (rsin(theta))^2 = 1
r^2cos^2(theta) + r^2sin^2(theta) = 1
r^2(cos^2(theta)) + r^2(sin^2(theta)) = 1
r^2=1
r= +- 1
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Reflection

We have been taking chapter test after chapter test and i have to say it's getting pretty boring. I want to learn something new. It's pretty crazy too how state is this weekend. To be honest, i'm really looking forward to it. I kinda need a break from life for a few days. I'm kinda looking forward to spring fest too. I hope this week is good and flies by.

relfection

EXAMPLE: y=x^3=4x

a.) x-axis
(-y)=x^3+4x
y=-x^3-4x Reflect; No

b.) y-axis
y=(-x^3)+4(-x)
y=-x^3-4x No

c.) y=x
x=y^3+4y
y^3+4y=x
y(y^2+4)=x No

d.) origin
(-y)=(-x)^3+4(-x)
-y=-x^3-4x
y=x^3+4x Yes



EXAMPLE: y=x^3+4x^2+12
For any type of polynomial the domain would be (-oo,oo) and for the range, odd:
(-oo,oo); quadratics: [vertex,oo] or [-oo,vertex].
The answer to this problem would be Domain: (-oo,oo) Range (-oo,oo)


EXAMPLE: 5x+4/x^2-4
First, set the bottom of the fraction equal to zero.
x^2-4=0
+4+4
x^2=4
x=+ or - 2
Your answer then comes to,
Domain: (-oo,-2)u(-2,2)u(2,oo)

reflection

Law of Sines: sin(opp. angle)/Leg=sin(Opp. angle)/Leg. Set up a proportion.

Example:
triangle ABC where A=36 degrees a=3 and B=56 degrees. find b

sin36/3=sin56/x 3sin56/sin36= x Plug into your calculator and done.

Law of Cosines:

(opposite leg)^2=(adjacent leg)^2+(other opposite leg)^2-2(leg)(leg)cos(angle in between)

example:
for a triangle with C=36 degrees a=5 b=6

c^2=5^2+6^2-2(5)(6)cos36
c=Square root of(25+36-2(5)(6)cos36)
c= 3.53

Area of non-right triangle:
1/2(leg)(leg)sin(angle between)

Example: Triangle ABC has sides a=5 b=3 and C=40 degrees

therefore your formula will be 1/2)(5)(3)(sin(40)) type that in your calculator and you will get the answer.

Reflection 3/21

I'm gunna review sequences for this reflection.....
If you have the sequence:
2, 5, 8, 11,........ it is arithmetic because it increases by 3 each time.
So the formula to find another term down the line would be:
t(n) = 2 + (n-1)3
because the formula for an arithmetic series is t(n) = t(1) + (n-1)d....t(1) means term one, n is the term number you are looking for, and d is the amount the series increases by.
-So if i wanted to find the 7th term of this sequence, if would plug in:
t(7) = 2 + (7-1)3which equals 21.
So the 7th term in this series is 21.
Now, the sequence 3, 4.5, 6.75,.... is a geometric sequence because it doesn't increase by the same number each time, it goes by a ratio in this case.
To find the ration, divide 4.5 by 3.....it equals 3/2. Then divide 6.75 by 4.5 jus to make sure...it equals 3/2. If it wouldn't have it would have been neither.
So now how do you find the 7th term in this sequence....by using this formula to start:
t(n) = t(1) x r^(n-1)-r is your ratio.
So lets plug in an get our answer:
t(7) = 3 x 3/2^(7-1)its gives you 2187/64, so thats your 7th term.

reflection march 21

This week was just another week, that went by sooooooooooo slow! We started off with some practice tests, and then we had our exam on Tuesday. Which was pretty easy to me and I think to everyone else too. Then we worked on more practice tests and I don't remember anything from. And it was Mr. St. Pierre's last week with us..I am definately not ready to go back to doing real work all the time:( I hope trig or whatever we are going to do when Mrs. Robinson gets back isn't too hard and I hope its easy to understand.

Reflection

I'll try explaining exponential equations i'm pretty sure I know how to do those.
It's a long process but it isn't too difficult.

in an equation:
(b^2/a)^-2 TIMES (a^2/b)^-3

you must first distribute your exponents to all parts of your fraction, in the case of exponents, you must multiply the exponential values together,

(b^-4/a^-2) TIMES (a^-6/b^-3)

in order to remove all negative exponents from the equation, put a 1 over each and use the sandwich method!

(1/b^4)/(1/a^2) TIMES (1/a^6)/(1/b^3)

multiply the top by the bottom(BREAD) and the two inner fractions by each other

a^2/b^4 TIMES b^3/a^6

cancel what you can:

a^2 and a^6= a^4
b^4 and b^3= b


FINAL EQUATION:

1/ba^4

Reflection 3/21

These are some of the things i understood from the beginning of the school year. I'm so glad that B-rob is coming back on mondayy! Now we can finally learn math again, and not read BOOKS! So i have no clue what we are going to learn when she comes back, but i hope it is really simple that i get it. I am going to do really good the last nine weeks. As long as i write my notes and do good on my quizzes then i will be alright.
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HOW TO FIND SYMMETRY:


y=x^2+1


1. x-axis:
(-y)=x^2+1
y=-x^2-1; not symmetric

2. y-axis:
y=(-x)^2+1
y=x^2+1; symmetric

3.y=x
x=y^2+1
y^2=x-1
y=[square root]x-1; not symmetric

4.(-y)=(-x)^2+1
-y=x^2+1
y=-x^2-1; not symmetric

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To find (f + g) (x)
It would convert in simpler terms to f(x) + g(x)

EXAMPLE:

f(x) = 2x + 1 g(x) = 2x^2 + 3
Find (f + g) (x)
f(x) + g(x)
(2x +1) + (2x^2 + 3)

(f + g) (x) = 2x^2 + 2x + 4



To find (f times g)(x)
It would convert in simpler terms to f(x) times g(x)

EXAMPLE:

f(x) = 4x + 4 g(x) = x - 2x^2
f(x) times g(x)
(4x + 4)(x - 2x^2)
Foil it out.
4x^2 -8x^3 + 4x -8x^2

-8x^3 - 4x^2 + 4x
(f times g) (x) = -8x^3 - 4x^2 + 4x



**if it says to find f(g(x)), you plug g(x) into every x in the f equation.

EXAMPLE:

f(x) = 2x + 1 g(x) = x + 4
f(g(x)) = 2(x + 4) + 1
f(g(x)) = 2x + 8 + 1
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Synthetic division, (easiest thing we've learned):


Equation: (x^5+x^4+2x^3+5x^2-3x+6) / (x+3)

First you'll set x+3=0
x=-3


-3|1 1 2 5 -3 6




1st: Bring your first number down.

-3|1 1 2 5 -3 6
------------------
1

2nd: multiply the number at the bottom by the number being divided and add.

-3|1 1 2 5 -3 6
| -3 6 -24 57 -162
---------------------------
1 -2 8 -19 54 -156

Remainder for this equation would be:
R= -156.

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**Some things that I am still iffy on are domain and range from the beginning of the year, if someone wants to explain how all of that works, please! and i also still don't understand how to do the word problems for trig. I could use some example problems, because they all deal with formulas and i'm not good with formulas, so please, examples? THANKS :) Can't wait to see B-rob!

Reflection

I found that these were sorta difficult. Although we didn't learn as much as normal, you had to try and understand more steps. This week we learned about domain, range, functions, finding any symmetry, and inverses. What I found was the easiest to learn was finding the symmetry.

EXAMPLE: y=x^3=4x

a.) x-axis
(-y)=x^3+4x
y=-x^3-4x Reflect; No

b.) y-axis
y=(-x^3)+4(-x)
y=-x^3-4x No

c.) y=x
x=y^3+4y
y^3+4y=x
y(y^2+4)=x No

d.) origin
(-y)=(-x)^3+4(-x)
-y=-x^3-4x
y=x^3+4x Yes

I also understood how to do domain and range very well. The types of the domain and range problems i understood the most were the fractions and polynomials. (oo => stands for infiniti

EXAMPLE: y=x^3+4x^2+12
For any type of polynomial the domain would be (-oo,oo) and for the range, odd:
(-oo,oo); quadratics: [vertex,oo] or [-oo,vertex].
The answer to this problem would be Domain: (-oo,oo) Range (-oo,oo)

EXAMPLE: 5x+4/x^2-4
First, set the bottom of the fraction equal to zero.
x^2-4=0
+4+4
x^2=4
x=+ or - 2
Your answer then comes to,
Domain: (-oo,-2)u(-2,2)u(2,oo)

What I am having the most trouble understanding is the inverses. I get most of the concept, but I'm still getting confused. Can anyone help me with that??