Dustin's Final Reflection

This year has been a tough year, but a good year. Although my grades weren't great, I learned alot. The major things we covered were trig and calc. I loved trig when it was just triangles, then this year I saw a different side of trig. It was difficult, but I understood it in the end. It will help me in Calculus and it will help me in college, so it was worth the hard work.

We also learned a lot of calculus towards the end of the year. That was the one thing that I just perfectly understood at the end of the year. This year was great and I loved my class. I could've done much better and I will remember to do that next year after seeing how my grades turned out.

Devin's Reflection

How to Find the Inverse of a Function:

• Replace f(x) with y
• Reverse the roles of x and y
• Solve for y in terms of x
• Replace y with f-1(x)
• check: should equal to x

Example:

f(x) = √x + 4

(x)^2 = (√y + 4)^2

x^2 = y + 4

y = x^2 - 4

f-1(x) = (x^2 - 4)

f(f-1(x)) = f(x^2 - 4) = √(x^2 - 4) + 4 = x

f-1(f(x)) = f-1(√x + 4) = (√x + 4)^2 - 4 = x + 4 - 4 = x

Logarithm Properties:

• logb MN = logb M + logb N
• logb M/N = logb M - logb N
• logb M^K = K logb M
• logb b^k = k
• b^logb^k = k

Changing Bases: (Done when you can't solve a log)

• Rewrite it as an exponential
• Take the log of both sides
• Move the variable to the front
• then solve

Example:

log5 10 = x

5^x = 10

log 5^x = log 10

x log 5 = 1

x = 1/log 5

Devin's Reflection

Completing the Square:

You can use completing the square to solve a quadratic equation when factoring doesn’t work. This method can only work when 1 is the coefficient of x².

For example:

x² + 6x - 2 = 0

x² + 6x = 2

x² + 6x + 9 = -2 + 9

(x + 3)² = 7

x + 3 = √7

x = -3 ± √7

(-3 + √7,0) (-3 -√7,0)

Rational Root therom:

Example: f(x)= 2x^3 + 3x^2 - 8 + 3

Step 1: find all possible roots..
p: factors of 3: 1, -1, 3, -3
q: factors of 2: 1, -1, 2, -2

*p is the leading constant term & q is the leading coefficient
possible roots are (p/q): 1, -1, 1/2, -1/2, 3, -3, 3/2, -3/2

Step 2: plug roots in calc & the zeros will be: 1, 1/2, -3

Step 3: synthetic division: (x - 1) (2x^2 + 5x + 3)

Step 4: slove further (factor): (x - 1) (2x^2 + 5x + 3)= (x - 1) (2x - 1) (x + 3)

Answer: x = 1, 1/2, -3

Domain & Range of functions:

Polynomials-domain of all polynomials is (−∞, ∞).

Fractions-you set the bottom to zero, solve for x, and then set up intervals

Square Roots-domain: set the inside = to zero, then set a # line, try values on either side of each #, and get ride of the negatives-range:graph

Absolute Value-domain: (- ∞ , + ∞)-range: [0 , + ∞)

Devin's Reflection

SOLVING TRIG EQUATIONS

for any line m = tan alpha
m = slope , alpha = angle of inclination
For a conic: tan 2 alpha = B/A-C
if A=C then pi/4
A = coefficient of x^2, B = coefficient of xy, C = coefficient of y^2

y=Asin(Bx-h)+C
amplitude is hight
b is period p=2π/b
h is horizontal shift
c is vertical shift

Identities:

1. check identities
2. algebra
3. identities

Reciprocal Relationships

• cscΘ=1/sinΘ
• secΘ=1/cosΘ
• cotΘ=1/tanΘ

Relationships with Negatives

• sin -Θ= -sinΘ and cos -Θ= -cosΘ
• csc -Θ= -cscΘ and sec -Θ= -secΘ
• tan -Θ= -tanΘ and cot -Θ= -cotΘ

Pythagorean Relationsihps

• sin²Θ+cos²Θ=1
• 1+tan²Θ=sec²Θ
• 1+cot²Θ=csc²Θ

Cofunction Relationships

• sinΘ=cos(90°-Θ) and cosΘ=sin(90°-Θ)
• tanΘ=cot(90°-Θ) and cotΘ=tan(90°-Θ)
• secΘ=csc(90°-Θ) and cscΘ=sec(90°-Θ)

Devin's Reflection

Reference Angles (must be between 0° and 90°)
1)find which quadrant angle is in
2)determine the sign in that quadrant (+ve or -ve)
3)subtract 180° until the angle is between 0° and 90° (0 and π/2)

1)find the reference angle using chart or calculator
2)find what quadrant you need to be in based on the sign of the value
3)use notes to move to that quadrant
To Move:
I to IV = make negative and add 360°
I to III = add 180°
I to II = make negative and add 180°
II to IV = add 180°

Unit Circle

sin theta = y/r

cos theta = x/r

tan theta = y/x

csc theta = r/x

sec theta = x/y

cot theta = x/y

r=sqrtx^2+y^2

90 pi/2
180 pi
270 3pi/2
360 2pi

sin pos in one and two and neg in three and four

cos pos in one and four and neg in two and three

tan pos in one and three and neg in two and four

cot pos in one and neg in two three and four

sec pos in one and neg in two three and four

csc pos in one and two and neg in three and four

AREA OF A NON-RIGHT TRIANGLE
A=1/2(leg)(leg)sin(angle between)

RIGHT TRIANGLES

• hypotenuse opposite angle
• a=1/2bh
• SOHCAHTOA

sin theta=opposite/hypotenuse
cos theta=adjacent/hypotenuse
tan theta=opposite/adjacent

LAW OF SINES
sinA/a sinB/b sinC/c

• used when you know pairs in a non-right triangle
• you are setting up proportions

Right Triangles

• A=1/2bh
• SOHCAHTOA
• sinΘ=opposite leg/hypotenuse
• cosΘ=adjacent leg/hypotenuse
• tanΘ=opposite leg/adjacent leg

Law of Cosines
(opposite leg)²=(adjacent leg)² + (other leg)² - 2(adjacent leg)(adjacent leg)cos°
EG: x, 5, 6 angle = 35°
x²=5²+6²-2(5)(6)cos35°
x=√(5²+6²-2(5)(6)cos35°)
x≈3.443

Area of Inscribed Shapes
A=nr²sinΘcosΘ

Devin's Reflection

Triangle trigonometry:
sine = opp/hyp
cos = adj/hyp
tan = opp/adj
csc = hyp/opp
sec=hyp/adj
cot=adj/opp

Moving between quadrants:

I to IV = make it negative and add 360º
I to III = add 180º
I to II = make it negative and add 180º

II to IV = move 180º

Law of sines (sinA/a) = (sinB/b) = (sinC/c)
Law of cosines (opp leg)² = (other adj leg)² -2(adj leg) (adj Leg) cos (angle b/w)

sinΘ = y/r
cosΘ = x/r

tanΘ = y/x
cotΘ = x/y
cscΘ = r/y

secΘ = r/x

r=√(x² + y²)

Graphing Trig functions:

y=Asin(Bx-h)+C

A = amplitude or height

B determines period(p)

p = (2π/B)

h = ((phase shift-horizontal shift)/(opposite))

C= vertical shift

Trigonometric Identites:
csc = 1/sin x
sec = 1/cos x
cot x = 1/tan x
sin (-x) = -sin x
cos (-x) = cos x
csc (-x) = -csc x
sec (-x) = sec x
tan (-x) = -tan x
cot (-x) = -cot x
sin^2 x+cos^2 x = 1
1+tan^2 x = sec^2 x
1+cot^2 x = csc^2 x
sin x = cos(90*-x)
tan x = cot (90*-x)
sec x = csc (90*-x)
cos x = sin (90*-x)
cot x = tan (90*-x)
csc x = sec (90*-x)
tan x = sinx/cosx
cot x = cosx/sinx

Devin's Reflection

Ch. 6 Conics

The standard equation of a circle is (x-h)^2+(y-k)^2 .....the center is (h,k)

To find the intersection of a line and a circle:

1. solve the linear eqn for y.
2. substitute in the circle eqn.
3. solve for x.
4. plug the x value in to get the y value.

If your x value is imaginary, then there is no point of intersection.

EX: find the center and radius.(x-3)^2+(y+7)^2=19

c:(h,k)

center: (3,-7)

radius: square root of 19

--Parabolas have no major axis and no asymptotes.

Axis of symmetry x=-b/2a

Finding the vertex
(-b/2a, f(-b/2a))

or

complete the square to get vertex form
y=(x+a)^2+b a&b are #'s

(-a,b) vertex

focus: 1/4p= coeff of x^2 then add p.

directrix is p units behind vertex. subtract p.

EX: 1/8y^2

v(0,0)

Focus: p=2(2,0)

directrix: x=-2

Devin's Reflection

CIRCLES
The equation of a circle in standard form is (x-h)^2-(y-h)^2=r^2 with the center being (h,k) and r being the radius.
Finding the intersection of a line and a circle:
1) solve linear equation for y
2) substitute in circle equation
3) solve for x
4) plug x in to get y value
(if x happens to be imaginary, there is no point of intersection)

ELLIPSES
1) (x-h)^2/(length of x/2)^2 + (y-k)^2/(length of y/2)^2 =1
2)center is (h,k)
3) major axis has larger denominator
4) vertex is on major axis
5) focus is smaller denom squared = larger denom squared - focus squared
focus is on major axis
Graphing:
1) find center
2) major axis = plus or minus the square root of the bigger denom
3) vertex
4) other intercepts
5) focus
6) length of major axis = 2 square root of
7) length of minor axis = 2 square root of
8) graph

HYPERBOLAS
1) (x+h)^2/(length/2)^2 - (y-k)^2/(length/2)^2 =1
OR
-(x-h)^2/(length/2)^2 + (y-k)^2/(length/2)^2 =1
2) center (h,k)
3) major axis is non-negative
4) vertex is the square root of non-negative denom
5) asymptotes y=+/-(square root of y)/(square root of x)x
6) focus^2 = x denom + y denom
focus^2 = vertex^2 + other denom

to sketch:
1) shape
2) center
3) major
4) minor
5) other intercept - none for hyperbolas
6) focus
7) asymptotes y=+/-square root of y/square root of x
8) vertex
9) sketch
A) draw a box using the vertex and +/-sr of other denom
B) draw diagonal through box corners
C) sketch a parabola on each vertex
D) label focus and asymptotes

Devin's Reflection

1.) area of a non right triangle = 1/2 (leg)(leg)SIN(angle b/w)

Example: non-right triangle: HIJ (left to right)H = 65 degrees, j = 2, i = 6. Find the area.

A = (1/2)(2)(6)sin(65)

A = 5.438

2.) Law of Sines(used to non-right triangles):

Sin A/a = Sin B/b= Sin C/c

Example: you have a triangle with the sides 4 and 5 & you also have an angle of 30 degrees.

A = 1/2 (4) (5) Sin 30 degrees

A = 10 Sin 30 degrees which is aproximately = 5

3.) Law of Cosines (used when you can't use Law of Sines):

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

Example: you have a triangle with the sides of 5, 6, and 7. find the angle between 5 and 6.

7^2=6^2+5^2-2(5)(6)

cos a7^2-6^2-5^2= 2(5)(6)

cos acos a= 7^2-6^2-5^2 / -2(6)(5)

a= cos-1 ((7^2-^6^2-5^2)/(-2(5)(6))

a= 78.463 degrees

4.) For any line : m = tan (alpha)
**m = slope , (alpha) = angle of inclination

5.) For a conic: tan 2 (alpha) = B/A-C
**if A=C then pie/4 (always)
**A = coefficient of x^2, B = coefficient of xy, C = coefficient of y^2

Examples:

1. Find the angle of inclination of x^2 - 2xy + 3y^2 = 1.

tan 2 (alpha) = B/A-C

A = 1 , B = -2 , C = 3

tan 2 (alpha) = -2/1 -3 = 1

tan 2 (alpha) = 1

2A = tan^-1 (1)

2 (alpha) = 45 , 225

alpha = 45/2 , 225/2

alpha = 22.5 , 112.5

2. x^2 + y^2 - 3xy + 4x - sqrt.

x = 1alpha = 1 (because A = 1 & C = 1 so A = C)

Devin's Reflection

**Logs

Condense:

Ex) logm + log7 + 4logn

= log7mn^4

Ex) 5loga + logd + log6

= log6da^5

Ex) 4logt - logc

= t^4/c

Ex) logn - 3logh -logy

= n/yh^3

Expand:

Ex) log5gh^2

= log5 + 2logh +logg

Ex) m^3b^7/f

= 3logm + 7logb - logf


**The Unit Circle

90 degrees, (0,1), pi/2

180 degrees, (-1,0), pi

270 degrees, (0,-1), 3pi/2

360 degrees, (1,0), 2pi


**6 Trig Functions

sin = y/r

cos = x/r

tan = y/x

csc = r/y

sec = r/x

cot = x/y


**Degrees & Radians

Degrees to radians= Degree * pi/180

Radians to degrees= Radians * 180/pi

**To solve coterminal angles, either add or subtract 360 to the angle.

Devin's Reflection

Simplifying Trig Function

1. Check identities

2. Algebra (factoring, combining like terms, and fraction)

3. Check identites

Some Proofs to help

-cotx= cosx/sinx

-tanx= sinx/cosx

-1+cot^2x=csc^2x

-1+tan^2x=sec^2x

sin^2+cos^2=1

The way to simplify is to find and rearrange the functions in a way the makes them resemble one of the proofs. After you have done that then you use the proofs to replace something in the equation. And then you keep repeating those steps until you can not do it anymore without making the equation bigger.

Ex. Prove sec^4x-tan^4x/sec^2x

(sec^2x-tan^2x)(sec^2x+tan^2x)/sec^2x

1=sec^2x-tan^2x

1(sec^2x+tan^2x)/sec^2x

sin^2x/cos^2x/1

=1+sin/2x

Devin's Final Reflection

This year has been filled with much knowledge and alot (ALOT) of difficulty. But this year I learned alot of information that I will much need in the next educational level. I learned trig and even though it was difficult, it will be a major help in calc and college. Even though trig was hard, it was not the hardest thing for me to learn this year. The section that gave me the most difficuly, was the section dealing with sequences.

There are 2 main types of sequences:

1.) Arithmetic- tn*t1+(n-1)d

n=term # t1=first term d=what you add

2.) Geometric- tn=t1*r^(n-1)

r= what you multiply by

Example: find the formula for the nth term of the arithmetic sequence

3,5,7

tn= 3+(n-1)(2)
tn=3+2n-2
tn=1+2n

Example: find the formula for the nth term of the sequence

3, 4.5, 6.75

divide the second term by the first to get your r.

4.5/3= 3/2 r= 3/2

tn=3(3/2)^n-1

Taylor final reflection

The major concepts of Advanced math were all the things dealing with trig
the focus and most stress of this year was learning and dealing with trig
all trig stems from the unit circle and the trig chart so ill include that

The Unit Circle

90 degrees, (0,1), pi/2

180 degrees, (-1,0), pi

270 degrees, (0,-1), 3pi/2

360 degrees, (1,0), 2pi


TRIGCHART
0° *sin 0= 0 *cos 0= 1 *tan 0= 0 *csc 0= undefined *sec 0= 1
*cot 0= undefined

30° * sin π/6= 1/2 *cos π/6= √3/2 *tan π/6= √3/3 *csc π/6= 2 *sec π /6= 2 √3/3
*cot π/6= √3
45° *sin π/4= √2/2 *cos π/4= √2/2 *tan π/4= 1 *csc π/4= √2 *sec π/4= √2
*cot π/4= 1
60° *sin π/3= √3/2 *cos π/3= 1/2 *tan π/3= √3 *csc π/3= 2 √3/3 *sec π/3= 2
*cot π/3= √3/2
90° *sin π/2= 1 * cos π/2= 0 *tan π/2= undefined *csc π/2= 1 *sec π/2= undefined *cot π/2= 0


i feel like this year really helped me with advancing my math score on the act
when i took the act last june i made a ninteen in math but this time when i took it in april i made a twenty five
i really did learn alot thus year


finally the one moment where something really clicked was when i missed the lesson on refrence anglwa
when mrs robinson broke down the lesson into three easy steps i got it easily

there are three steps to finding a refrence angle
STEPS
pre step- set up ____ trig function____
the rest will plug into this
#1- discover which quadrent the given angle is in
#2- determine if the given trig function ((which will go in the "trig function" spot)) is positive or negative in that quadrent ((this will go into the first blank))
#3- change given angle to some number between 0 and 90 degrees by subtracting 180 until the angle is between 0 and 90 degrees. ((the number that you get which is between 0 and 90 degrees will go in the second blank))

ALAINA'S FINAL REFLECTION

This year we have covered a lot of material. B-rob has helped and has been a great teacher. The class wasn't "easy" but it wasn't "hard" either. I struggled on some concepts but i eventually understood them.
the major concepts were:
review on algebra 2
solving polynomials
inequalities
domain and range
exponents
conics
trigonometry
trig with triangles
solving trig equations
trig formulas
polar
sequences and series

What did I gain?
As this course is coming to an end, I feel as if I've gained a mutual knowledge of the subject. I learned how to graph conics and actually understood it and used trig to build a bridge.

Methods that helped me:
Well first off, B-rob is a great teacher. She's very visual and explanitory. She gives a lot of examples of each type of whatever it is we're learning. Also, I have to work some things on my own in order to understand them.

Final Blog!

Okay, so I think that one of the most major concepts that we covered this year was knowing the Trig Identites:
csc = 1/sin x
sec = 1/cos x
cot x = 1/tan x
sin (-x) = -sin x
cos (-x) = cos x
csc (-x) = -csc x
sec (-x) = sec x
tan (-x) = -tan x
cot (-x) = -cot x
sin^2 x+cos^2 x = 1
1+tan^2 x = sec^2 x
1+cot^2 x = csc^2 x
sin x = cos(90*-x)
tan x = cot (90*-x)
sec x = csc (90*-x)
cos x = sin (90*-x)
cot x = tan (90*-x)
csc x = sec (90*-x)
tan x = sinx/cosx
cot x = cosx/sinx

To me this and the trig chart were two of the most important things we learned because you had to memorize all these identities, and without them you would not be able to do a lot of the problems in trig, and memorizing the chart helps you simplify an answer to the simplest form possible.

An activity that really helped me this year was when B-rob told me that the whole trig chart can be derived by just knowing the first few lines. sin cos and tan's answers can be flipped and that gives you the csc sec and cot's answers.

Devin's Make up

Exponents:

1. b^x * b^y = b^x + y....example: 2^3 * 2^5 = 2^8

2. b^x/b^y = b^x - y....example: 5^7/5^4 = 5^3

3. (ab)^x = a^xb^x....example: (3 * 7)^3 = 3^3 * 7^3

4. (a/b)^x = a^x/b^x....example: (3/5)^3 = 3^3/5^3

5. (b^x)^y = b^xy....example: (2^2)^3 = 2^6

6. b^x/y = y^√b^x....examples: 5^3/4 = 3^√5^3

7. to solve for exponents:

• write as the same base
• set exponents equal
• then solve for x

examples:

(a). 5^3x = 5^7x - 2

Examples:
log3^9=2
In order to solve, you need to put this into exponential form...
3^2=9
...and that is your answer.

log2^16=4
2^4=16

log5^125=x
5^x=125 (if it asks for exponential form)
x=3 (if it says to solve)

log x=2
10^2=x
x=100

logx^8=3
x^3=8
x=2

Devin's Make up

When it is b^x*b^y= b^x+y. When it s b^x/b^y= b^x-y. When it is (ab)^x= a^xb^y. When it is (a/b)^x= a^x/b^x. When it is (b^x)^y= b^xy. When it is b^x/y= y{b^x}.

To solve for an exponent
a. write as the same base
b. set exponents equal
c. solve for x

Simplfy
(b^2/a) ^-2

-b^-4/a^-2

-1/b^41/a^2

= a^2/b^4

With double fraction, you have to multiply the outsides by each other, and the insides byeach other.

Ex. (a^-2+b^-2)^-1

-(1/a^2+1/b^2)^-1

-(b^2/b^2*1/a^2+1/b^2*a^2/a^2)^-1

-(b^2/a^2b^2+a^2/a^2b^2)^-1

-(b^2+a^2/a^2b^2)^-1

=a^2b^2/b^2+a^2

This week we also covered logirythms (I doubt I spelled that correctly).
logb x=a

- b^a=x

Ex. log2 8=x

-2^x=8

x= 3

Domain of logs and ln = (0,infinity)
Range of logs and ln = (-infinity, infinity)

Devin's Make up

Logarithm Properties:

• logb MN = logb M + logb N
• logb M/N = logb M - logb N
• logb M^K = K logb M
• logb b^k = k (this one i don't get..maybe i copied it wrong)
• b^logb^k = k

Here are some examples:

1. log 2 + log 3 + log 4 = log 24 (mulitply: 2 x 3 x 4)

2. log 8 + log 5 - log 4 = log 10 (mulitply: 8 x 5 then divide: 40/4)

3. 2 ln 6 - ln 3 = ln 12 (raise 6 to the 2nd power = 36 the divided by 3 = 12)

4. log M - 3 log N = log M/ N^3

5. ln 2 + ln 6 - 1/2 ln 9 = ln 12/3 = ln 4

6. Expand logb MN^2....logb M + 2 logb N

7. Condense log 45 - 2 log 3....log (45/9) = log 5

8. Rewrite in exponetial form: log36 6 = 1/2....36^1/2 = 6

9. Rewrite in logarithmic form: 2^2 = 4....log2 4 = 2

Changing Bases: (Done when you can't solve a log)

• Rewrite it as an exponential
• Take the log of both sides
• Move the variable to the front
• then solve

(use the same steps when solving for x as an exponent when you can't write them as the same base)

examples:

1. log5 10 = x

5^x = 10

log 5^x = log 10

x log 5 = 1

x = 1/log 5

2. 2^x = 7

log 2^x = log 7

x log 2 = log 7

x = log 7/log 2

Devin's Make up

How to change bases:
1)rewrite problem in exponential form
2)take the log of both sides
3)move the variable to the front
4)solve
Eg. log5of10=x
5^x=10
log5x=log10
xlog5=1
x=1/log5

Graphing exponential functions (ab^x):
if b is greater than 1, the graph goes up whereas if it is less than one, the graph goes down
Eg. f(x)=5(3)^x the graph would go up because b is greater than one

The formulas are pretty ease but remembering them will be pretty hard and using them in the correct problem will also be kinda hard.

A(t)=Ao(1+r)^t
Ao = what you start with
r = rate
t = time

A(t)=Aob^t/k
Ao = what you start with
t = time
b = double, half, etc.
k = regular time to double, half, etc.
Eg. half-life of 5 days
b = ½
k = 5
A(t)=Ao(1/2)^t/5

P(t)=Poe^rt
Po = wha tyou start with
r = rate
t = time
you only use the problem with p when it says compounding continuously in the problem

then theres the limit thing and rule of 72 (72 / r% = how long it takes to double

Devin's Make up

Exponential functions

All you have to do is basically plug things in and then ur set. There are 3 different formulas:
A(t)=Ao(l+r)^t
A(t)=Aob^t/k
Ao=what you start with
b=double, half, etc (use this if u see double, half, etc in the problem)
k=time reg. to double, half, etc
t=time
P(t)=Poe^rt
Po=what you start with
r=rate
t=time
(only use this when compounding continuously)

All you basically have to do is plug the numbers in and solve if it says so.

The steps for condensing logs are easy
First you have to remember the relations
Mn = m+n
m/n = m-n
m^k = k log M
sub b B^k = k
b^log sub b^k = K

so any problem will fit into one of these relations
Ex: expand log sub b MN^2
Log sub b M + 2 Log sub b N