My brilliant wall by Sally Thomas

Unit Circle

zthierfelder19 — 2017-11-08 18:46:51 UTC

- angle measure (from x-axis to terminal side) is angle measure in radians
- y-value is sine of that angle
- x-value is cosine of that angle
- y/x is tangent

Graphing sine

zthierfelder19 — 2017-11-08 18:56:21 UTC

- period: 2pi
- amplitude: 1
- crosses through origin halfway through the uphill portion
- some important points: (0,0), (pi/2, 1), (pi, 0), (3pi.2, -1), (2pi, 0)

Graphing cosine

zthierfelder19 — 2017-11-08 19:04:17 UTC

- period: 2pi
- amplitude: 1
- does not pass through origin, crosses y-axis at y=1 at the highest point in the period
- some important points: (0,1), (pi/2, 0), (pi, -1), (3pi/2, 0), (2pi, 1)
- unit circle for cosine is the sine unit circle rotated a quarter turn to the left (to help you remember the points)

Quadratics (Unite 1)

smazzarella19 — 2017-11-08 19:06:21 UTC

DISCIMINATES
- Use discriminate (value under square root in quad formula) to determine the number of solutions
- Discriminate > 0 then there are 2 solutinos
- Discriminate = 0 then there is 1 solution
- Discriminate < 0 then there are 0 solutions
Quardratic Vertex
- (-b)/ 2a
Interval Notation
- [ x ] = including x
- ( x ) = exlcuding x
Extrema
- Top of peak/ bottom of valley
  - Local extrema is a top/ bottom of a zoomed in portion of the graph
  - Absolute extrema is the bottom/ top of a graph
Symmetry
- Even (y-axis)
  - f(x)= f(-x)
- Odd (x- axis)
  - f(x) = -f(-x)
Limit Notation
- just learn it!!!!

6 Trig Functions

zthierfelder19 — 2017-11-08 19:10:53 UTC

- SOH CAH TOA
- sinx = opposite/hypotenuse
- cosx = adjacent/hypotenuse
- tanx = opposite/adjacent
- cscx = hypotenuse/opposite (reciprocal of sine)
- secx = hypotenuse/adjacent (reciprocal of cosine)
- cotx = adjacent/hypotenuse (reciprocal of tangent)

Graphing tangent

zthierfelder19 — 2017-11-08 19:16:43 UTC

- period: pi
-asymptotes at every interval of pi/2
- passes through origin

Graphing cosecant

zthierfelder19 — 2017-11-08 19:18:07 UTC

- inverse of sinx
- period: 2pi
- asymptotes at every interval of pi
- range: (-infinity, -1]. [1, infinity)

Graphing secant

zthierfelder19 — 2017-11-08 19:20:29 UTC

- inverse of cosx
- period: 2pi
- asymptotes at every interval of pi/2
- range: (-infinity, -1], [1, infinity)

Graphing cotangent

zthierfelder19 — 2017-11-08 19:32:33 UTC

- inverse of tanx
- period: pi
- asymptotes every interval of pi

Piecewise Functions

sthomas192 — 2017-11-08 19:34:10 UTC

definition: a function that is divided into multiple intervals with a different function rule applied to each interval

Composition of Functions

sthomas192 — 2017-11-08 19:34:59 UTC

Let f and g be two functions, where the domain of f is the range of g. The composition of f of g is
(f * g)(x)=f (g(x))

Using period to solve functions

zthierfelder19 — 2017-11-08 19:35:16 UTC

Go to the nearest complete period of the function and evaluate the remainder for the value.
EX: sin 432843628463982743pi
period of 2pi so nearest complete period is 432843628463982742pi, remainder of pi
sin(pi) = 0 so...
sin 432843628463982743pi = 0

Algebraic Combination Rules

sthomas192 — 2017-11-08 19:41:10 UTC

(f + g)(x) = f(x) + g(x)
(f - g)(x) = f(x) - g(x)
(fg)(x) = f(x)g(x)
(f/g)(x) = f(x)/g(x)

Angle terminology

zthierfelder19 — 2017-11-08 19:43:35 UTC

- pi radians = 180º
- sector length = theta x radians
- positive angle is a counter-clockwise turn
- negative angle is a clockwise turn
- coterminal angles: angles that share the same initial and terminal sides

Decomposing Functions

sthomas192 — 2017-11-08 20:21:06 UTC

finding functions f and g such that h(x) = f(g)(x)

Relation

sthomas192 — 2017-11-08 20:21:57 UTC

- definition: a set of ordered pairs (x,y)

12 Basic Functions

sthomas192 — 2017-11-08 20:29:17 UTC

f(x) = x: the identity function
f(x) = x^2: the squaring function
f(x) = x^3: the cubing function
f(x) = 1/x: the reciprocal function
f(x) = SQRT(x): the square foot function
f(x) = e^x: the exponential function
f(x) = ln x: the natural logarithm function
f(x) = sin x: the sine function
f(x) = cos x: the cosine function
f(x) = abs(x): the absolute value function
f(x) = int(x): the greatest integer function
f(x) = 1/(1+e^-x): the logistic function

Ahoy there!

sthomas192 — 2017-11-08 20:34:54 UTC

Dear all,
My name is Sally Thomas, and I am so excited to be here to contribute to this brilliant wall. I am a boarder living in the McClellan house from Rumson, New Jersey. I look forward to hearing everyone's ideas, so let's get learning! :) I <3 knowledge!

Heyoooo

kmcalister20 — 2017-11-08 20:56:41 UTC

My name is Kelsey McAlister. Contrary to popular belief, I am a sophomore and a day student. I'm a part of StanFam<333 I play ice hockey so I'm kinda #tough #swole #feisty. I identify as a Minnesotan even though I've lived in dirty jerz my whole life. Here's a pic of me when I used to think I could model :) happy learning!

Inverses

kmcalister20 — 2017-11-08 21:07:33 UTC

Steps for finding an inverse algebraically:
1) Find a one-to-one function
2) Switch the x and y
3) Solve for y
4) Find the new domain and it's restrictions

Finding an inverse graphically:
1) Reflect the graph over the line y=x
**Hint: The points (a, b) and (b, a) are reflections of eachother over y=x

Finding the Domain of an Inverse Function:
Domain of Inverse = Range of Orginal

How to verify an inverse function:
1) Solve for f(g(x)) and then g(f(x)), if they both equal x then it is an inverse function.

Graphical Transformations

kmcalister20 — 2017-11-08 21:12:15 UTC

Reflections:
-f(x) is across the x-axis
f(-x) is across the y-axis
-f(-x) is through the origin
Shifts:
Vertical shift by d when
y= f(x) + d
Horizontal shift by -c when
y= (f(x) + c)
Stretch/shrink
Vertical stretch when
y=k(f(x)) when k>1
Vertical shrink when
y=k(f(x)) when 0(If k is negative then follow same format except you also have to reflect)
Horizontal stretch when
y=f(B(x)) 1/B is 0Horizontal shrink when
y=f(B(x)) 1/B is B>1

If y=f(x) has points (x,y) then y= A*f(B(x-c))+D has points ((1/B)x-c, Ay+D)
A: Vertical stretch/shrink
B: Horrizontal stretch/shrink
C: Horizontal shift
D: Vertical shift

***MAKE SURE TO MAKE THE COEFFICIENT OF X 1!!
Do not leave y=f(3x-6)
Change it to: y=f(3(x-2))

Modeling with Functions

kmcalister20 — 2017-11-08 21:28:00 UTC

Write the area A of a circle as a function of:
Its radius r:
A=pi(r)^2
Its diameter d:
A=pi(d/2)^2
Its circumference C:
A=pi(c/2pi)^2

Greeting friends

zthierfelder19 — 2017-11-09 03:14:49 UTC

Let's take a journey. -- Snoop Dog (i'm pretty sure he was referring to the beautiful journey of learning precalculus which I am blessed to be taking with all of you)
I'm Zoe Thierfelder and am a #daystud from princeton nj and a member of the lovely McClellan house along with my mcsister sally thomas! A fun fact about me: in 6th grade I told everyone I was a universalist because I didn't have a favorite food, color, or animal. Yikes.

Welcome

smazzarella19 — 2017-11-09 14:03:28 UTC

Hi there, welcome to My Brilliant Wall. I am one the brilliant administrators Simona! Unlike any of the other administrators, I don't play any sports #freewoman. I live in a faraway land called Beijing, but when on campus I live in STEFUNZ #besttodatesince88 Fun fact, Stephens is better than McCllelan and Stanley!! Fun fact again I'm a coxswain. My last name is also kinda cheesy

Horizontal Line Test

kmcalister20 — 2017-11-09 14:04:34 UTC

If you can draw horizontal lines across a graph and each line only touches 1 unique y-value then it's inverse is a function.

Unit Analysis

zthierfelder19 — 2017-11-09 14:07:40 UTC

When you have a unit/unit and need it in another unit/another unit...
1) map out how to get from one unit to the other (ie what are the steps you would take to get from min/feet to hours/miles)
2) if you have a unit on the top that you're trying to cancel out, put it on the bottom of the number you're multiplying it by (so if you have min/feet and are trying to into hours, do min/feet x 1 hour/60 min). if the number is on the bottom, put it on the top.
3) once you have your units completely mapped out, plug the numbers into your calculator.

sthomas192 — 2017-11-09 14:18:19 UTC

Quadratic Modeling and Equation Solving

smazzarella19 — 2017-11-09 14:20:29 UTC

quadratic equation
Completing the Square
Graphing it
- oops you won't need this because it's a NO calculator section

kmcalister20 — 2017-11-09 14:21:29 UTC

Study to change your life!

sthomas192 — 2017-11-09 14:21:56 UTC

zthierfelder19 — 2017-11-09 14:22:33 UTC

zthierfelder19 — 2017-11-09 14:24:05 UTC

can't see or touch precalc but i sure think it's beautiful

sthomas192 — 2017-11-09 14:24:07 UTC

sthomas192 — 2017-11-09 14:24:13 UTC

sthomas192 — 2017-11-09 14:24:17 UTC

kmcalister20 — 2017-11-09 14:25:02 UTC

STUDY TO CHANGE YOUR LIFE!

smazzarella19 — 2017-11-09 14:26:05 UTC

kmcalister20 — 2017-11-09 14:26:35 UTC

Only re-read previous chapters when you're studying for finals!!!! YOU CAN'T GO ONTO THE NEXT CHAPTER OF YOUR LIFE IF YOUR FINAL GRADE SUCKS SO STUDY STUDY STUDY!!

sthomas192 — 2017-11-09 14:26:35 UTC

PRECALC IS THE FUTURE

zthierfelder19 — 2017-11-09 14:27:02 UTC

stay cool my guys

sthomas192 — 2017-11-09 14:30:24 UTC

zthierfelder19 — 2017-11-09 14:31:05 UTC

smazzarella19 — 2017-11-09 14:32:09 UTC

sthomas192 — 2017-11-09 14:32:22 UTC

zthierfelder19 — 2017-11-09 14:32:33 UTC

kmcalister20 — 2017-11-09 14:32:44 UTC

kmcalister20 — 2017-11-09 14:33:12 UTC

Just because you fail the first time doesn't mean you stop trying!

smazzarella19 — 2017-11-09 14:34:10 UTC

smazzarella19 — 2017-11-09 14:36:12 UTC

Functions

smazzarella19 — 2017-11-09 14:38:08 UTC

Domain
- input; horizontal axis; x
Range
- output; vertical axis; y
There is only ONE domain for every range

Graphical Transformations of Sinusoids

zthierfelder19 — 2017-11-09 14:43:45 UTC