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# Math Study Sheet!!!!
# Essential Skills (1)
> ## Simple Arithmetics
>> ### Addition / Subtraction
>>> | Expression | Equivalent|
>>> |:----------:|:---------:|
>>> | a + b | a + b |
>>> | (-a) + b | b - a |
>>> | a + (-b) | a - b |
>>> | (-a) + (-b) | -(a + b) |
>>> | a - b | a - b|
>>> | a - (-b) | a + b |
>>> | (-a) -(-b) | (-a) + b|
>> ### Multiplication / Division
>>> | Signs | Outcome |
>>> |:-----:|:-------:|
>>> | a * b | Positive |
>>> | (-a) * b | Negative |
>>> | a * (-b) | Negative |
>>> | (-a) * (-b) | Positive |
>> ### BEDMAS / PEMDAS
>>> Follow ```BEDMAS``` for order of operations if there are more than one operation
>>> | Letter | Meaning |
>>> |:------:|:-------:|
>>> | B / P | Bracket / Parentheses |
>>> | E | Exponent |
>>> | D | Divison |
>>> | M | Multiplication |
>>> | A | Addition |
>>> | S | Subtraction |
>>> <img src="https://ecdn.teacherspayteachers.com/thumbitem/Order-of-Operations-PEMDAS-Poster-3032619-1500876016/original-3032619-1.jpg" width="300">
> ## Interval Notation
>> A notation that represents an interval as a pair of numbers.
>> The numbers in the interval represent the endpoint. E.g. **[x > 3, x &isin; R]**
>> ```|``` means ```such that```
>> ```E``` or &isin; means ```element of```
>> ```W``` represents **Whole Numbers** (W = {x | x > 0, x &isin; Z})
>> ```N``` represents **Natural Numbers** (N = {x | x &ge; 0, x &isin; Z})
>> ```Z``` represents **Integers** (Z = {x | -&infin; &le; x &ge; &infin;, x &isin; Z})
>> ```Q``` represents **Rational Numbers (Q = {<sup>a</sup>&frasl;<sub>b</sub> |a, b &isin; Z, b &ne; 0})
>> | Symbol | Meaning |
>> |:------:|:-------:|
>> | (a, b) | Between but not including ```a``` or ```b```, you also use this for ```∞``` |
>> | [a, b] | Inclusive |
>> | a ∪ b | Union (or) |
>> | a ∩ b | Intersection (and) |
> ## Pythgorean Theorem
>> let a be the adjecant and b be the opposite, and c be the hypotenuse.
>> a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>
>> <img src="https://docs.google.com/drawings/u/1/d/sGjyHDIs-wHWzppHAGdIpEA/image?w=162&h=70&rev=1&ac=1&parent=1ZIXKcDk3LBlgPK2EoUV04c0G1LZotrtfgVhJTooO1zA" width="200">
> ## Operations with Rationals
>> Q = { <sup>a</sup>&frasl;<sub>b</sub> | a, b &isin; Z, b &ne; 0 }
>> Any operations with rationals, there are 2 sets of rules
>>> 1. ```Rules for operations with integers```
>>> 2. ```Rules for operations with fractions```
>> To Add / subtract rationals, find common denominator and then add / subtract numerator
>> To Multiply rationals, first reduce the fraction to their lowest terms, then multiply numerators and denominators
>> To Divide rationals, multiply them by the reciprocal
>> ### Example Simplify Fully:
>>> = <sup>3</sup>&frasl;<sub>4</sub> &divide; <sup>2</sup>&frasl;<sub>12</sub> [Reduce to lowest terms]
>>> = <sup>3</sup>&frasl;<sub>4</sub> &divide; <sup>1</sup>&frasl;<sub>7</sub> [Multiply by reciprocal]
>>> = <sup>3</sup>&frasl;<sub>4</sub> &times; 7
>>> = <sup>21</sup>&frasl;<sub>4</sub> [Leave as an improper fraction]
>> ### Shortcut for multiplying fractions
>>> cross divide to keep your numbers small
>>> Example:
>>> <sup>2</sup>&frasl;<sub>4</sub> &times; <sup>4</sup>&frasl;<sub>6</sub>
>>> <sup>1</sup>&frasl;<sub>1</sub> &times; <sup>1</sup>&frasl;<sub>3</sub>
>>> = <sup>1</sup>&frasl;<sub>3</sub>
>> ## Exponent Laws
>>> | Rule | Description| Example |
>>> |:----:|:----------:|:-------:|
>>> |Product|a<sup>m</sup> &times; a<sup>n</sup> = a<sup>n+m</sup>|2<sup>3</sup> &times; 2<sup>2</sup> = 2<sup>5</sup>|
>>> |Quotient|a<sup>m</sup> &divide; a<sup>n</sup> = a<sup>n-m</sup>|3<sup>4</sup> &divide; 3<sup>2</sup> = 3<sup>2</sup>|
>>> |Power of a Power|(a<sup>m</sup>)<sup>n</sup> = a<sup>mn</sup>|(2<sup>3</sup>)<sup>2</sup> = 2<sup>6</sup>|
>>> |Power of a Quotient|(<sup>a</sup>&frasl;<sub>b</sub>)<sup>n</sup> = <sup>a<sup>n</sup></sup>&frasl;<sub>b</sub><sup>n</sup>|(<sup>2</sup>&frasl;<sub>3</sub>)<sup>4</sup> = <sup>2<sup>4</sup></sup>&frasl;(<sub>3</sub><sup>4</sup>)|
>>> |Zero as Exponents|a<sup>0</sup> = 1|21<sup>0</sup> = 1|
>>> |Negative Exponents|a<sup>-m</sup> = <sup>1</sup>&frasl;<sub>a</sub><sup>m</sup>|1<sup>-10</sup> = <sup>1</sup>&frasl;(<sub>1</sub><sup>10</sup>) or <sup>1</sup>&frasl;<sub>1</sub>|
>>> |Rational Exponents|a<sup>n/m</sup> = (<sup>m</sup>&radic;a)<sup>n</sup>|16<sup>5/4</sup> = (<sup>4</sup>&radic;16)<sup>5</sup> = 2<sup>5</sup>|
>>> **Note:**
>>> Standard --> Expanded Form
>>> 6<sup>4</sup> = 6 &times; 6 &times; 6 &times; 6
>> ## Scientific Notation
>>> They convey accuracy and precision. It can either be written as its original number or in scientific notation:
>>> 555 (**Exact**) or 5.55 x 10<sup>2</sup> (**3 significant figures**).
>>> In scientific notation, values are written in the form **a(10<sup>n</sup>)**, where ```a``` is a number within 1 and 10 and ```n``` is any integer.
>>> Some examples include the following: 5.4 x 10<sup>3</sup>, 3.0 x 10<sup>2</sup>, and 4.56 x 10<sup>-4</sup>.
>>> When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used
>>> <img src="https://embedwistia-a.akamaihd.net/deliveries/d2de1eb00bafe7ca3a2d00349db23a4117a8f3b8.jpg?image_crop_resized=960x600" width="500">
>> ## Rates, Ratio and Percent
>>> ```Ratio```: A comparison of quantities with the same unit. These are to be reduced to lowest terms.
>>> Examples: ```a:b, a:b:c, a/b, a to b ```
>>> ```Rates```: A comparison of quantities expressed in different units.
>>> Example: ```10km/hour```
>>> ```Percent```: A fraction or ratio in which the denominator is 100
>>> Examples: ```50%, 240/100```
> ## Number Lines
>> a line that goes from a point to another point, a way to visualize set notations and the like
>> <img src="https://i2.wp.com/mathblog.wpengine.com/wp-content/uploads/2017/03/numberlines-thumbnail.jpeg?resize=573%2C247&ssl=1" width="500">
>> A solid filled dot is used for ```[]``` and a empty dot is used for ```()```
> ## Tips
>> Watch out for the ```+/-``` signs
>> Make sure to review your knowledge of the exponent laws
>> For scientific notation, watch out for the decimal point
>> Use shortcut when multiplying fractions
# Polyomials (2)
> ## Introduction to Polynomials
>> A ```variable``` is a letter that represents one or more numbers
>> An ```algebraic expression``` is a combination of variables and constants ```(e.g. x+y+6 = 5)```
>> When a specific value is assigned to a variable in a algebraic expression, this is known as substitution.
> ## Methods to solve a polynomial
>> 1. ```Combine like terms```
>> 2. ```Dividing polynomials```
>> 3. ```Multiplying polynomials```
> ## Simplifying Alegebraic Expressions
>> An algebraic expression is an expression with numbers, variables, and operations. You may expand or simplify equations thereon.
> ## Factoring
>>Two methods of solving; decomposition and criss-cross. First of all, the polynomial must be in the form of a quadratic
>> equation (ax<sup>2</sup> + bx + c). As well, simplify the polynomial, so that all common factors are outside
>> (e.g 5x + 10 = 5(x + 2) ).
>> |Type of Polynomial|Definition|
>> |:-----------------|:---------|
>> |Monomial|Polynomial that only has one term|
>> |Binomial|Polynomial that only has 2 terms|
>> |Trinomial|polynomial that only has 3 terms|
>> |Type|Example|
>> |:--:|:-----:|
>> |Perfect Square Trinomials| (a+b)<sup>2</sup> = a<sup>2</sup>+2ab+b<sup>2</sup> or (a-b)<sup>2</sup> = a<sup>2</sup>-2ab+b<sup></sup>|
>> |Difference with Squares|a<sup>2</sup>-b<sup>2</sup> = (a+b)(a-b)|
>> |Simple Trinomials|x<sup>2</sup>+6x-7 = (x+7)(x-1)|
>> |Complex Trinomials|2x<sup>2</sup>-21x-11 = (2x+1)(x-11)|
>> |Common Factor|2ab+6b+4 = 2(ab+3b+2)|
>> |Factor By Grouping|ax+ay+bx+by = (ax+ay)+(bx+by) = a(x+y)+b(x+y) = (a+b)(x+y)|
> ## Shortcuts
>> <img src="https://image.slidesharecdn.com/factoringquadraticexpressions-120625145841-phpapp01/95/factoring-quadratic-expressions-13-728.jpg?cb=1340636365" width="500">
> ## Foil / Rainbow Method
>> <img src = "https://calcworkshop.com/wp-content/uploads/foil-method-formula.png" width ="500">
> ## Definitions
>> ```Term``` a variable that may have coefficient(s) or a constant
>> ```Alebraic Expressions```: made up of one or more terms
>> ```Like-terms```: same variables raised to the same exponent
> ## Tips
>> Be sure to factor fully
>> Learn the ```criss-cross``` (not mandatory but its a really good method to factor quadratics)
>> Learn ```long division``` (not mandatory but its a really good method to find factors of an expression)
>> Remember your formulas
>> Simplify first, combine like terms
# Solving Equations and Inequailties (3)
> ## Equations
>> a ```mathematical statement``` in which the value on the ```left side``` equals the value on the ```right side``` of the equal sign
>> To ```solve``` and equation is to find the variable that makes the statement true
>> ### Methods to solve an equation
>>> 1. Expand and simplify both sides
>>> 2. Isolate using reverse order of operations
>>> 3. Check the solution by plugging the variable back into the equation and check if the ```left side``` equals the ```right side```
> ## Venn Diagrams
>> <img src="https://d2slcw3kip6qmk.cloudfront.net/marketing/blog/2017Q3/Venn-diagram-symbols-and-notation/VDIntersections.png" width="400">
>> ```Set```: a collection of elements, O (the circle)
>> ```Universal Set```: This is a collection of all the elements that you are interested in. Use ```{}``` bracket to write the set inside the rectangle
>> ```Union```: &cup;
>> if 2 sets have union, all the elements belong to any of the set. This is known as ```or```
>> ```Intersection```: &cap;
>> if 2 sets have an intersection, they are elements belonging to both sets. This is known as ```and```
>> ```Set Notation```:
>> A notation that represents the collection of numbers. It is written in this form x = {x|x&isin;R}
>> ```Absolute Value```
>>> There are 2 cases. For this sort of equation, you must split the equation into 2 separate equations. One of the
>>> equations will have the absolute bracket be positive while the other negative.
>>> Absolute values are written in the form ```| x |```
>>> where
>>> if x > 0, | x | = x
>>> if x == 0, | x | = 0
>>> if x < 0, | x | = -x
> ## Quadractic Equations
>> ```Quadratic Function```: A parabolic graph where the axis of symmetry is parallel to the y-axis
>> ```Quadratic Equation```: This function is set equal to ```0```. The solution to the equation are called ```roots```
>> Solve quadratic equation by:
>> 1. Isolation
>> a(x+b)<sup>2</sup> + k = 0
>> 2. Factor using zero-product property
>> ```The Zero Factor Property``` refers to when a&times;b=0, then either a=0 or b=0.
>> (x-a)(x-b)=0
>> x = a, b
>> <img src="http://www.assignmentpoint.com/wp-content/uploads/2017/12/Quadratic-Expression-1.jpg" width="400">
>> Note:
>> &radic;x<sup>2</sup> = &plusmn; x (There are 2 possible solutions)
>> ```Distrubutive Property``` - This is opening the bracket. a(x+y) = ax+ay
>> ## Discriminant
>>> The discriminant determines the number of solutions (roots) there are in a quadratic equation. ```a```, ```b```, ```c``` are the
>>> coefficients and constant of a quadratic equation: ```y = ax<sup>2</sup> + bx + c```
>>> D = b<sup>2</sup> - 4ac
>>> D > 0 ```(2 distinct real solutions)```
>>> D = 0 ```(1 real solution)```
>>> D < 0 ```(no real solutions)```
>>> <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">
> ## Solving Linear-Quadratic Systems
>> To find the point of intersection, do the following:
>> 1. Isolate both equations for ```y```
>> 2. Set the equations equal to each other by ```subsitution``` Equation 1 = Equation 2
>> 3. Simplify and put everything on one side and equal to zero on the other side
>> 4. Factor
>> 5. Use zero-product property to solve for all possible x-values
>> 6. Subsitute the x-values to one of the original equations to solve for all y-values
>> 7. State a conclusion / the solution
> <img src = "https://lh5.googleusercontent.com/AJxSjT24kwneM_UH6kehfX-7AnzVewTJIk6v02aXOZ84veou2xNyBMPmhGSXWNhvhJfZT-wwHSlDNvbsfeHzjpGSuXMOohoIvaS2u0saoO1BZTRV3xNVobdoWytLhkVl0CkEaIiQ" width ="500">
> There are 3 possible cases
> In addition, to determine the number of solutions, you the Discriminant formula **D = b<sup>2</sup> - 4ac**
> ## Tips
>> ```Absolute Values``` can have 2 solutions
>> ```Quadratics``` can also have 2 solutions
>> Make sure to do the reverse when moving things to the other side, meaning a positive on the ```left side``` becomes a negative on the ```right side```
# Measurement and Geometry (4)
> ## Angle Theorems
> 1. ```Transversal Parallel Line Theorems``` (TPT)
> a. Alternate Angles are Equal ```(Z-Pattern)```
> b. Corresponding Angles Equal ```(F-Pattern)```
> c. Interior Angles add up to 180 ```(C-Pattern)```
> <img src="https://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/58a52a99-05da-4595-88b8-2cbca91e8bbf.gif" width="300">
> 2. ```Supplementary Angle Triangle``` (SAT)
> - When two angles add up to 180 degrees
> <img src="https://embedwistia-a.akamaihd.net/deliveries/cdd1e2ebe803fc21144cfd933984eafe2c0fb935.jpg?image_crop_resized=960x600" width="500">
> 3. ```Opposite Angle Theorem (OAT)``` (OAT)
> - Two lines intersect, two angles form opposite. They have equal measures
> <img src="https://images.slideplayer.com/18/6174952/slides/slide_2.jpg" width="400">
> 4. ```Complementary Angle Theorem``` (CAT)
> - The sum of two angles that add up to 90 degrees
> <img src="https://images.tutorvista.com/cms/images/67/complementary-angle.png" width="300">
> 5. ```Angle Sum of a Triangle Theorem``` (ASTT)
> - The sum of the three interior angles of any triangle is 180 degrees
> <img src="https://dj1hlxw0wr920.cloudfront.net/userfiles/wyzfiles/f0516fa1-669b-441d-9f11-a33907a2a0b0.gif" width="300">
> 6. ```Exterior Angle Theorem``` (EAT)
> - The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles
> <img src="https://www.katesmathlessons.com/uploads/1/6/1/0/1610286/exterior-angle-theorem-diagram-picture_orig.png" width="300">
> 7. ``` Isosceles Triangle Theorem``` (ITT)
> - The base angles in any isosceles triangle are equal
> <img src="http://www.assignmentpoint.com/wp-content/uploads/2016/06/isosceles-triangle-theorem.jpg" width="400">
> ## Properties of Quadrilaterals
>> Determine the shape using the properties of it
>> |Figure|Properties|
>> |:-----|:---------|
>> |Scalene Triangle|no sides equal|Length of line segment|
>> |Isosceles Triangle| two sides equal|Length of line segment|
>> |Equilateral Triangle|All sides equal|Length of line segment|
>> |Right Angle Triangle|Two sides are perpendicular to each other|
>> |Parallelogram|Opposite sides are parallel and have equal length. Additionally, the diagonals bisect each other|
>> |Rectangle|Adjacent sides are perpendicular to each other. Furthermore, the diagonals bisect each other and are equal in length|
>> |Square|All sides are equal in length. The adjacent sides and diagonals are perpendicular. The adjacent sides are equal in length, so as the diagonals|
>> |Rhombus|Opposite sides are parallel and all sides are equal to each other, the diagonals are perpendicular|
>> |Trapezoid|There is one pair of opposite sides and they are parallel and unequal in length|
>> |Kite|The diagonals are perpendicular|
> ## 2D Geometry Equations
>> |Shape|Formula|Picture|
>> |:----|:------|:------|
>> |Rectangle|```Area```: lw <br> ```Perimeter```: 2(l+w)|<img src="https://lh5.googleusercontent.com/pJ-Pm3oA8_oLRtc_eEhTBrGqXazzsK4phUQ_QqrNl5vyk1t6pypO9waRBonuHJYv4_Z2I9G2jq8xc-6Tmhy3MpihW9ITWna49JQNDtQ" width="200">|
>> |Triangle|```Area```: bh/2 <br> ```Perimeter```: a+b+c|<img src="https://lh6.googleusercontent.com/covvHwXxQhrK2Hr0YZoivPkHodgstVUpAQcjpg8sIKU25iquSHrRd2EJT64iWLsg_75WnBw4T9P0OTBiZDkpqEkXxflZQrL16sNhcFfet_z4Mw5EPFgdx_4HzsagV0Sm5jN6EKr_" width="200">|
>> |Circle|```Area```: πr<sup>2</sup> <br> ```Circumference```: 2πr or πd|<img src="https://lh5.googleusercontent.com/RydffLVrOKuXPDXO0WGPpb93R8Ucm27qaQXuxNy_fdEcLmuGZH4eYc1ILNmLEx8_EYrRuOuxFavtL9DF1lTWYOx9WaYauVlu0o_UR6eZLeGewGjFNUQSK8ie4eTm1BMHfRoQWHob" width="200">|
>> |Trapezoid|```Area```: (a+b)h/2 <br> ```Perimeter```: a+b+c+d|<img src="https://lh6.googleusercontent.com/_nceVtFlScBbup6-sPMulUTV3MMKu1nonei0D1WY-KRkpHSbPCIWgDO8UGDQBGKh8i0dkAqOhFUHl7YHCFOt6AMRSJiXALlBBY0mBo1MMZxHRVcg8DknSlv4ng7_QswcZtaRwrJb" width="200">|
> ## 3D Geometry Equations
>> |3D Object|Formula|Picture|
>> |:----|:------|:------|
>> |Rectangular Prism|```Volume```: lwh <br> ```SA```: 2(lw+lh+wh)|<img src="https://lh6.googleusercontent.com/-mqEJ4AMk3xDPfqH5kdVukhtCGl3fgTy2ojyAArla54c7UoAnqKW_bsYSaFySXLplE59pqLIg5ANZAL1f6UEejsrKJwQCfyO7gwUQmSDoJJtQG_WkfHcOFDjidXV4Y4jfU2iA5b-" width="200">|
>> |Square Based Pyramid|```Volume```: <sup>1</sup>&frasl;<sub>3</sub>b<sup>2</sup>h <br> ```SA```: 2bs+b<sup>2</sup>|<img src="https://lh3.googleusercontent.com/BFePwVkXcQA-dFlBD8DQQ5Aq_fQ0PQLm6LIhMN27IAMjlWax0s_t5pcfv6HEy1JghOE8bQtCf90KZ4CEcLbicBlAFx3KOlNJ4XecSdFx" width="200">|
>> |Sphere|```Volume```: <sup>4</sup>&frasl;<sub>3</sub>πr<sup>3</sup> <br> ```SA```: 4πr<sup>2</sup>|<img src="https://lh6.googleusercontent.com/DL6ViJLbap2dcSAlZnYKR7c33033g8WuJVvqz0KpzCyIJ0wXyrh5ejoLhrTxlX9uASQlxPmihm8doU1sNbaQxqBcTaPnP-lC6LUrPqzPNi11AHiHQAu3ag7uIGcwzkdC9e5uo1en" width="200">|
>> |Cone|```Volume```: <sup>1</sup>&frasl;<sub>3</sub>πr<sup>2</sup>h <br> ```SA```: πrs+πr<sup>2</sup>|<img src="https://lh5.googleusercontent.com/V3iZzX8ARcipdJiPPFYso_il3v_tcrYHZlFnq3qkekRSVBVcj8OzWxMuBqN45aHbv6y-fDH4uY11Gus3KMrvf_Z_TvsfJCwZZ19Ezf7Yj6DzVirp-Gx3V0Qy793ooUwTDmdKW_xq" width="200">|
>> |Cylinder|```Volume```: πr<sup>2</sup>h <br> ```SA```: 2πr<sup>2</sup>+2πh|<img src="https://lh5.googleusercontent.com/4uWukD3oNUYBG-fLX2-g58X8at0h74al7BJI5l78LZ0Bu9nXuZnt9dp9xiETeLTqykP-WWFdO_H5by4RkgDVxSENZgootSrAsOUoY2RWubflNOAau1bVFgm9YIe59SmiFlyxwgDV" width="200">|
>> |Triangular Prism|```Volume```: ah+bh+ch+bl <br> ```SA```: <sup>1</sup>&frasl;<sub>2</sub>blh|<img src="https://lh3.googleusercontent.com/_oRUVgfdksfUXGKQk3AtrtY70E8jEq-RRK-lB9yKc_Rtio2f2utGAY-rI4UqjWEeTzUoN_r7EiqdZZeeE12EY-fiV55QQKdvnv4y4VaxQ9xt9Izugp6Ox_LqIUpQzPKVldptgKWm" width="200">|
> ## Optimization (For Maximimizing Area/Volume, or Minimizing Perimeter/Surface Area)
>> ### 2D Objects
>> |Shape|Maximum Area|Minimum Perimeter|
>> |:----|:-----------|:----------------|
>> |4-sided rectangle|A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width<br>A = lw<br>A<sub>max</sub> = (w)(w)<br>A<sub>max</sub> = w<sup>2</sup>|A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width.<br>P = 2(l+w)<br>P<sub>min</sub> = 2(w)(w)<br>P<sub>min</sub> = 2(2w)<br>P<sub>min</sub> = 4w|
>> |3-sided rectangle|l = 2w<br>A = lw<br>A<sub>max</sub> = 2w(w)<br>A<sub>max</sub> = 2w<sup>2</sup>|l = 2w<br>P = l+w2<br>P<sub>min</sub> = 2w+2w<br>P<sub>min</sub> = 4w|
>> ### 3D Objects
>> |3D Object|Maximum Volumne|Minimum Surface Area|
>> |:--------|:--------------|:-------------------|
>> |Cylinder(closed-top)|The cylinder must be similar to a cube where h = 2r<br>V = πr<sup>2</sup>h<br>V<sub>max</sub> = πr<sup>2</sup>(2r)<br>V<sub>max</sub> = 2πr<sup>2</sup>|The cylinder must be similar to a cube where h = 2r<br>SA = 2πr<sup>2</sup>+2πrh<br>SA<sub>min</sub> = 2πr<sup>2</sup>+2πr(2r)<br>SA<sub>min</sub> = 2πr<sup>2</sup>+4πr<sup>2</sup><br>SA<sub>min</sub> = 6πr<sup>2</sup>|
>> |Rectangular Prism(closed-top)|The prism must be a cube, <br> where l = w = h<br>V = lwh<br>V<sub>max</sub> = (w)(w)(w)<br>V<sub>max</sub> = w<sup>3</sup>|The prism must be a cube, <br>where l = w = h<br>SA = 2lh+2lw+2wh<br>SA<sub>min</sub> = 2w<sup>2</sup>+2w<sup>2</sup>+2w<sup>2</sup><br>SA<sub>min</sub> = 6w<sup>2</sup>|
>> |Cylinder(open-top)|h = r<br>V = πr<sup>2</sup>h<br>V<sub>max</sub> = πr<sup>2</sup>(r)<br>V<sub>max</sub> = πr<sup>3</sup>|h = r<br>SA = πr<sup>2</sup>+2πrh<br>SA<sub>min</sub> = πr<sup>2</sup>+2πr(r)<br>SA<sub>min</sub> = πr<sup>2</sup>+2πr<sup>2</sup><br>SA<sub>min</sub> = 3πr<sup>2</sup>|
>> |Square-Based Rectangular Prism(open-top)|h = w/2<br>V = lwh<br>V<sub>max</sub> = (w)(w)(<sup>w</sup>&frasl;<sub>2</sub>)<br>V<sub>max</sub> = <sup>w<sup>3</sup></sup>&frasl;<sub>2</sub>|h = w/2<br>SA = w<sup>2</sup>+4wh<br>SA<sub>min</sub> = w<sup>2</sup>+4w(<sup>w</sup>&frasl;<sub>2</sub>)<br>SA<sub>min</sub> = w<sup>2</sup>+2w<sup>2</sup><br>SA<sub>min</sub> = 3w<sup>2</sup>|
> ## Labelling
>> Given any polygons, labelling the vertices must always:
>> 1. use ```CAPITAL LETTERS```
>> 2. they have to be labeled in ```clockwise``` or ```counter-clockwise``` directions
>> For a triangle, the side lengths are labeled in ```LOWERCASE LETTERS``` associated to the opposite side of the vertex
>> <img src="http://www.technologyuk.net/mathematics/trigonometry/images/trigonometry_0073.gif" width="400">
> ## Median
>> Each median divides the triangle into 2 smaller triangles of equal area
>> The centroid is exactly <sup>2</sup>&frasl;<sub>3</sub> they way of each median from the vertex, or <sup>1</sup>&frasl;<sub>3</sub> the way from the midpoint of the opposite side, or ```2:1``` ratio
>> The three medians divide the triangle into ```6``` smaller triangles of equal area and ```3 pairs``` of congruent triangles
>> <img src="https://blog.udemy.com/wp-content/uploads/2014/05/d-median.png" width="500">
> ## Terms:
>> ```Altitude``` The height of a triangle, a line segment through a vertex and perpendicular to the opposite side
>> ```Orthocenter```: where all 3 altitudes of the triangle intersect
>>> <img src="https://mathbitsnotebook.com/Geometry/Constructions/orthocenter1a.jpg" width="300">
>> ```Midpoint```: A point on a line where the length of either side of the point are equal
>> ```Median```: A line segment joining the vertex to the midpoint of the opposite side
>> ```Midsegment```: A line joining 2 midpoints of the 2 sides of a triangle
>> ```Centroid```: The intersection of the 3 medians of a triangle
>>> <img src="http://www.mathwords.com/c/c_assets/centroid.jpg" width="300">
> ## Proportionality theorem:
>> The midsegment of a triangle is ```half``` the length of the opposite side and ```parallel``` to the opposite side
>> Three midsegment of a triangle divide ```4 congruent``` triangles with the same area
>> The Ratio of the outer triangle to the triangle created by the 3 midsegments is ```4 to 1```
>> <img src="https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcSPYlT2JwH4oMYHmpq0DLhBTi1goY0JaRBpNdmZBWgWKSaXAJTM" width="300">
> ## Tips
>> Make sure to know your optimization formualas
>> Read the word problems carefully, determine which formual to use
>> Never **ASSUME**, be sure to **CALCULATE** as most of the time the drawings are **NOT ACCURATE**
>> To find ```missing area```, take what you have, subtract what you don't want
>> Don't be afraid to draw lines to help you solve the problem
# Analytical Geometry and Linear Relations (5)
> ```Linear Relation```: A relation which a single straight line can be drawn through every data point and the first differences are constant
> ```Non - Linear Relation```: A single smooth curve can be drawn through every data point and the first differences are not constant
> ## Slope and Equation of Line
>> ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```change in y / change in x```
>> ```Slope Formula```: **m = y<sub>2</sub>-y<sub>1</sub>/x<sub>2</sub>-x<sub>1</sub>**
>> ```Standard Form```: **ax + by + c = 0**, a&isin;Z, b&isin;Z, c&isin;Z (must be integers and ```a``` must be positive)
>> ```Y-intercept Form```: **y = mx + b**
>> ```Point-slope Form```: **y<sub>2</sub>-y<sub>1</sub> = m(x<sub>2</sub>-x<sub>1</sub>)**
>> The slope of a vertical lines is undefined
>> The sloope of a horizontal line is 0
>> Parallel lines have the ```same slope```
>> Perpendicular slopes are negative reciprocals
> ## Relations
>> A relation can be described using
>> 1. Table of Values (see below)
>> 2. Equations (y = 3x + 5)
>> 3. Graphs (Graphing the equation)
>> 4. Words
>> When digging into the earth, the temperature rises according to the
>> following linear equation: t = 15 + 0.01 h. **t** is the increase in temperature in
>> degrees and **h** is the depth in meters.
> ## Perpendicular Bisector
>> To find the perpendicular bisector, you will need to fidn the slope and midpoint
>> Formula: slope1 &times; slope2 = -1
>> Notation: m<sub>&perp;</sub>
>> <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/parallel-perpendicular-lines/parallel_perpendicular_lines_1.gif" width="300">
> ## Definitions
>> ```Parallel```: 2 lines with the same slope
>> ```Perpendicular```: 2 lines with slopes that are the negative reciprocal to the other. They form a 90 degree angle where they meet.
>> ```Domain```: The **ordered** set of all possible values of the independent variable (x).
>> ```Range```: The **ordered** set of all possible values of the dependent variable (y).
>> ```Continous Data```: A data set that can be broken into smaller parts. This is represented by a ```Solid line```.
>> ```Discrete Data```: A data set that **cannot** be broken into smaller parts. This is represented by a ```Dashed line```.
>> ```First Difference```: the difference between 2 consecutive y values in a table of values which the difference between the x-values are constant.
>> ```Collinear Points```: points that line on the same straight line
> ## Variables
>> ```Independent Variable```: A Variable in a relation which the values can be chosen or isn't affected by anything.
>> ```Dependent Varaible```: A Variable in a relation which is **dependent** on the independent variable.
> ## Statistics
>> ```Interpolation```: Data **inside** the given data set range.
>> ```Extrapolation```: Data **outside** the data set range.
>> ```Line of Best Fit```: A line that goes through as many points as possible, and the points are the closest on either side of the line,
>> and it represents the trend of a graph.
>> ```Coefficient of Correlation```: The value that indicates the strength of two variables in a relation. 1 is the strongest and 0 is the weakest.
>> ```Partial Variation```: A Variation that represents a relation in which one variable is a multiple of the other plus a costant term.
> ## Time - Distance Graph
>> Time is the independent variable and distance is the dependent variable
>> You can't go backwards on the x-axis, as you can't go back in time
>> Plot the points accordingly
>> Draw the lines accordingly
>> <img src="https://dryuc24b85zbr.cloudfront.net/tes/resources/6061038/image?width=500&height=500&version=1519313844425" width="400">
> ## Scatterplot and Line of Best Fit
>> A scatterplot graph is there to show the relation between two variables in a table of values.
>> A line of best fit is a straight line that describes the relation between two variables.
>> If you are drawing a line of best fit, try to use as many data points, have an equal amount of points onto and under the line of best fit, and keep it as a straight line.
>> <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/line%20of%20best%20fit-eyeball/lineofbestfit-e-1.gif" width="300">
> ## Table of values
>> To find first differences or any points on the line, you can use a ```table of values```
>>| y | x |First Difference|
>>|:--|:--|:---------------|
>>|-1|-2|1|
>>|0|-1|1|
>>|1|0|1|
>>|2|1|1|
>>|3|2|1|
>>|4|3|1|
> ## Tips
>> Label your graph correctly, the scales/scaling and always the ```independent variable``` on the ```x-axis``` and the ```dependent variable``` on ```y-axis```
>> Draw your ```Line of Best Fit``` correctly
>> Read the word problems carefully, and make sure you understand it when graphing things
>> Sometimes its better not to draw the shape, as it might cloud your judgement (personal exprience)
>> Label your lines
# System of Equations (6)
> ## Linear System
>> Two or more equation that you are working on all together at once on the same set of axes.
>> The lines may ```cross``` or ```intersect``` at a point called the ```Point of Intersection (POI)```.
>> The coordinated of the ```POI``` must satisfy the equation of all the lines in a linear equation.
>> In business, the ```Point of Intersection``` is known as the **Break Even Point** where ```Revenue - Cost = Profit```
>> when **Profit = 0**. There is no gain or loss.
>> ### Number of Solutions
>>> <img src="https://lh5.googleusercontent.com/wqYggWjMVXvWdY9DiCFYGI7XSL4fXdiHsoZFkiXcDcE93JgZHzPkWSoZ6f4thJ-aLgKd0cvKJutG6_gmmStSpkVPJPOyvMF4-JcfS_hVRTdfuypJ0sD50tNf0n1rukcLBNqOv42A" width="500">
> # Ways to solve Systems of Equations
> 1. Subsitution
> Here we eliminate a variable by subbing in another variable from another equation
> We usually do this method if a variable is easily isolated
> Example:
> - ```
> y = x + 10 (1)
> x + y + 34 = 40 (2)
> ```
> We can sub (1) into (2) to find ```x```, then you the value of ```x``` we found to solve for ```y```
> ```x + (x + 10) + 34 = 40```
> ```2x + 44 = 40```
> ```2x = -4```
> ```x = -2```
> Then solve for ```y```
> ```y = -2 + 10```
> ```y = -8```
> 2. Elimination
> Here we eliminate a variable by basically eliminate a variable from an equation
> We usually use this method first when the variables are not easily isolated, then use subsitution to solve
> Example:
> - ```
> 2x + 3y = 10 (1)
> 4x + 3y = 14 (2)
> ```
> We can then use elimination
> ```
> 4x + 3y = 14
> 2x + 3y = 10
> ------------
> 2x + 0 = 4
> x = 2
> ```
> Then sub the value of ```x``` into an original equation and solve for ```y```
> ```2(2) + 3y = 10```
> ```3y = 6```
> ```y = 2```
> 3. Graphing
> we can rewrite the equations into ```y-intercept form``` and then graph the lines, and see where the lines intersect (P.O.I), and the P.O.I is the solution
> ## Solving Systems of Linear Inequalities
>> Find the intersection region as the ```solution```.
>> ## If:
>> | |Use ```Dash``` line|Use ```Solid line```|
>> |:-|:------------------|:-------------------|
>> |Shade the region ```above``` the line|y > mx + b|y &ge; mx + b|
>> |Shade the region ```below``` the line|y < mx + b| y &le; mx + b|
>> ## If
>> |x > a <br> x &ge; a|
>> |:------------------|
>> shade the region on the **right**
>> ## If
>> |x < a <br> x &le; a|
>> |:------------------|
>> shade the region on the **left**
>> Step 1. change all inequalities to ```y-intercept form```
>> Step 2. graph the line
>> Step 3. shade the region where all the regions overlap
>> <img src="https://www.mathplanet.com/Oldsite/media/29728/coordinateplane21.png" width="400">
> ## Tips
>> Read the questions carefully and model the system of equations correctly
>> Be sure to name your equations
>> Label your lines
# General Tips
> Be sure to watch out for units, like ```cm``` or ```km```
> Watch out for ```+/-```
> Be sure to reverse the operation when moving things to the other side of the equation
> Make sure to have a proper scale for graphs
> Read question carefully and use the appropriate tools to solve
> **WATCH OUT FOR CARELESS MISTAKES!!!!!!!!!!!**
> ## Word Problems
>> Read carefully
>> model equations correctly
>> ```Reread``` the question over and over again until you fully understand it and made sure there is no tricks. :p
> ## Graph Problems
>> Look up on tips in units (5) and (6)
>> be sure to use a ruler when graphing
> ## System of Equations
>> When in doubt or to check your work, just plug the numbers back in and check if the statement is true
# Credits
> Ryan Mark - He helped provide alot of information for me
> Magicalsoup - ME!