diff --git a/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md b/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md index 201e18a..78cfebe 100644 --- a/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md +++ b/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md @@ -1,637 +1,643 @@ # Math Study Sheet!!!! # Exam Detail -> |Unit|Marks| -> |:---|:----| -> |Unit 1|10| -> |Unit 2|10| -> |Unit 3|9| -> |Unit 4|11| -> |Unit 5|11| -> |Unit 6|8| -> |Forms|4| -> |Total|63| +|Unit|Marks| +|:---|:----| +|Unit 1|10| +|Unit 2|10| +|Unit 3|9| +|Unit 4|11| +|Unit 5|11| +|Unit 6|8| +|Forms|4| +|Total|63| -> |Section|Marks| -> |:------|:----| -> |Knowledge|21| -> |Application|23| -> |Thinking|12| -> |Communication|3| -> |Forms|4| +|Section|Marks| + |:------|:----| + |Knowledge|21| + |Application|23| + |Thinking|12| + |Communication|3| + |Forms|4| -> |Part|Question| -> |:---|:-------| -> |A|9 multiple choice| -> |B|10 Short Answer -->
- 7 Knowledge questions
- 3 Application Questions| -> |C|10 Open Response -->
- 10 Knowledge Questions
- 5 Application Questions
- 3 Thinking Questions
- 1 Communication Question| + |Part|Question| + |:---|:-------| + |A|9 multiple choice| + |B|10 Short Answer -->
- 7 Knowledge questions
- 3 Application Questions| + |C|10 Open Response -->
- 10 Knowledge Questions
- 5 Application Questions
- 3 Thinking Questions
- 1 Communication Question| -# 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| +# Unit 1: Essential Skills ->> ### 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 | - ->>> - -> ## 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 ∈ R]** ->> ```|``` means ```such that``` ->> ```E``` or ∈ means ```element of``` ->> ```N``` represents **Natural Numbers** (N = {x | x > 0, x ∈ Z}) ->> ```W``` represents **Whole Numbers** (W = {x | x ≥ 0, x ∈ Z}) ->> ```Z``` represents **Integers** (Z = {x | -∞ ≤ x ≤ ∞, x ∈ Z}) ->> ```Q``` represents **Rational Numbers** (Q = {a⁄b |a, b ∈ Z, b ≠ 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 ->> a and b are the two legs of the triangle or two sides that form a 90 degree angle of the triangle, c is the hypotenuse ->> a2 + b2 = c2 - ->> +## Simple Arithmetics -> ## Operations with Rationals ->> Q = { | a, b ∈ Z, b ≠ 0 } ->> ->> Any operations with rationals, there are 2 sets of rules ->>> 1. ```Rules for operations with integers``` ->>> 2. ```Rules for operations with fractions``` +### 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| ->> 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 +### Multiplication / Division + | Signs | Outcome | + |:-----:|:-------:| + | a * b | Positive | + | (-a) * b | Negative | + | a * (-b) | Negative | + | (-a) * (-b) | Positive | ->> ### Example Simplify Fully: ->>> [Reduce to lowest terms] +### BEDMAS / PEMDAS +- Follow ```BEDMAS``` for order of operations if there are more than one operation ->>> [Multiply by reciprocal] + | Letter | Meaning | + |:------:|:-------:| + | B / P | Bracket / Parentheses | + | E | Exponent | + | D | Divison | + | M | Multiplication | + | A | Addition | + | S | Subtraction | ->>> +- ->>> [Leave as an improper fraction] +## 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 ∈ R]** +- ```|``` means ```such that``` +- ```E``` or ∈ means ```element of``` +- ```N``` represents **Natural Numbers** (N = {x | x > 0, x ∈ Z}) +- ```W``` represents **Whole Numbers** (W = {x | x ≥ 0, x ∈ Z}) +- ```Z``` represents **Integers** (Z = {x | -∞ ≤ x ≤ ∞, x ∈ Z}) +- ```Q``` represents **Rational Numbers** (Q = {a⁄b |a, b ∈ Z, b ≠ 0}) ->> ### Shortcut for multiplying fractions ->>> cross divide to keep your numbers small ->>> Example: ->>> ->>> ->>> + | 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) | ->> ## Exponent Laws +## Pythgorean Theorem +- a and b are the two legs of the triangle or two sides that form a 90 degree angle of the triangle, c is the hypotenuse +- a2 + b2 = c2 ->>> | Rule | Description| Example | ->>> |:----:|:----------:|:-------:| ->>> |Product|am × an = an+m|23 × 22 = 25| ->>> |Quotient|am ÷ an = an-m|34 ÷ 32 = 32| ->>> |Power of a Power|(am)n = amn|(23)2 = 26| ->>> |Power of a Quotient| = | = | ->>> |Zero as Exponents|a0 = 1|210 = 1| ->>> |Negative Exponents|a-m = |1-10 = | ->>> |Rational Exponents|an/m = | = | +- + +## Operations with Rationals +- Q = { | a, b ∈ Z, b ≠ 0 } + +- Any operations with rationals, there are 2 sets of rules + 1. ```Rules for operations with integers``` + 2. ```Rules for operations with fractions``` ->>> **Note:** ->>> Exponential Form --> Expanded Form ->>> 64 = 6 × 6 × 6 × 6 +- 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 ->> ## 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 102 (**3 significant figures**). ->>> In scientific notation, values are written in the form **a(10n)**, where ```a``` is a number within 1 and 10 and ```n``` is any integer. ->>> Some examples include the following: 5.4 x 103, 3.0 x 102, and 4.56 x 10-4. ->>> When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used +### Example Simplify Fully: +- [Reduce to lowest terms] ->>> +- [Multiply by reciprocal] ->>> **Remember**: For scientific notation, round to ```3 significant``` digits +- ->> ## 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 ``` +- [Leave as an improper fraction] ->>> ```Rates```: A comparison of quantities expressed in different units. ->>> Example: ```10km/hour``` +### Shortcut for multiplying fractions +- cross divide to keep your numbers small +- Example: +- +- +- ->>> ```Percent```: A fraction or ratio in which the denominator is 100 ->>> Examples: ```50%, 240/100``` +## Exponent Laws -> ## Number Lines ->> a line that goes from a point to another point, a way to visualize set notations and the like ->> ->> A solid filled dot is used for ```[]``` and a empty dot is used for ```()``` + | Rule | Description| Example | + |:----:|:----------:|:-------:| + |Product|am × an = an+m|23 × 22 = 25| + |Quotient|am ÷ an = an-m|34 ÷ 32 = 32| + |Power of a Power|(am)n = amn|(23)2 = 26| + |Power of a Quotient| = | = | + |Zero as Exponents|a0 = 1|210 = 1| + |Negative Exponents|a-m = |1-10 = | + |Rational Exponents|an/m = | = | + +**Note:** +- Exponential Form --> Expanded Form +- 64 = 6 × 6 × 6 × 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 102 (**3 significant figures**). +- In scientific notation, values are written in the form **a(10n)**, where ```a``` is a number within 1 and 10 and ```n``` is any integer. +- Some examples include the following: 5.4 x 103, 3.0 x 102, and 4.56 x 10-4. +- When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used + +- + +- **Remember**: For scientific notation, round to ```3 significant``` digits + +## 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 +- +- 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 +## 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. y + 8)``` ->> 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``` +# Unit 2: Polyomials +## 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. y + 8)``` +- 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. +## 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 (ax2 + bx + c). As well, simplify the polynomial, so that all common factors are outside ->> (e.g 5x + 10 = 5(x + 2) ). +## Factoring +- Two methods of solving; decomposition and criss-cross. First of all, the polynomial must be in the form of a quadratic +- equation (ax2 + 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 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)2 = a2+2ab+b2 or (a-b)2 = a2-2ab+b| ->> |Difference with Squares|a2-b2 = (a+b)(a-b)| ->> |Simple Trinomials|x2+6x-7 = (x+7)(x-1)| ->> |Complex Trinomials|2x2-21x-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)| + |Type|Example| + |:--:|:-----:| + |Perfect Square Trinomials| (a+b)2 = a2+2ab+b2 or (a-b)2 = a2-2ab+b| + |Difference with Squares|a2-b2 = (a+b)(a-b)| + |Simple Trinomials|x2+6x-7 = (x+7)(x-1)| + |Complex Trinomials|2x2-21x-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 +## Shortcuts ->> +- -> ## Foil / Rainbow Method ->> +## Foil / Rainbow Method +- -> ## 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 +## 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 +## 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``` +# Unit 3: Solving Equations and Inequailties + +## 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``` -> -> ## Absolute Values ->> 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 +## Absolute Values +- 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)2 + k = 0 ->> 2. Factor using zero-product property ->> ```The Zero Factor Property``` refers to when a×b=0, then either a=0 or b=0. ->> (x-a)(x-b)=0 ->> x = a, b +## 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)2 + k = 0 + 2. Factor using zero-product property + - ```The Zero Factor Property``` refers to when a×b=0, then either a=0 or b=0. + - (x-a)(x-b)=0 + - x = a, b ->> Note: ->> √x2 = ± x (There are 2 possible solutions) ->> ```Distrubutive Property``` - This is opening the bracket. a(x+y) = ax+ay +- + +**Note:** +- √x2 = ± x (There are 2 possible solutions) +- ```Distrubutive Property``` - This is opening the bracket. a(x+y) = ax+ay -> ## 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``` +## 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)``` +# Unit 4: Measurement and Geometry +## 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)``` > -> 2. ```Supplementary Angle Triangle``` (SAT) -> - When two angles add up to 180 degrees +2. ```Supplementary Angle Triangle``` (SAT) + - When two angles add up to 180 degrees -> + - -> 3. ```Opposite Angle Theorem (OAT)``` (OAT) -> - Two lines intersect, two angles form opposite. They have equal measures +3. ```Opposite Angle Theorem (OAT)``` (OAT) + - Two lines intersect, two angles form opposite. They have equal measures -> + - -> 4. ```Complementary Angle Theorem``` (CAT) -> - The sum of two angles that add up to 90 degrees +4. ```Complementary Angle Theorem``` (CAT) + - The sum of two angles that add up to 90 degrees -> + - -> 5. ```Angle Sum of a Triangle Theorem``` (ASTT) -> - The sum of the three interior angles of any triangle is 180 degrees +5. ```Angle Sum of a Triangle Theorem``` (ASTT) + - The sum of the three interior angles of any triangle is 180 degrees -> + - -> 6. ```Exterior Angle Theorem``` (EAT) -> - The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles +6. ```Exterior Angle Theorem``` (EAT) + - The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles -> + - -> 7. ``` Isosceles Triangle Theorem``` (ITT) -> - The base angles in any isosceles triangle are equal +7. ``` Isosceles Triangle Theorem``` (ITT) + - The base angles in any isosceles triangle are equal -> + - -> 8. ```Sum of The Interior Angle of a Polygon``` -> - The sum of the interioir angles of any polygon is ```180(n-2)``` or ```180n - 360```, where ```n``` is the number of sides of the polygon +8. ```Sum of The Interior Angle of a Polygon``` + - The sum of the interioir angles of any polygon is ```180(n-2)``` or ```180n - 360```, where ```n``` is the number of sides of the polygon -> + - -> 9. ```Exterior Angles of a Convex Polygon``` -> - The sum of the exterior angle of any convex polygon is always ```360 degrees``` +9. ```Exterior Angles of a Convex Polygon``` + - The sum of the exterior angle of any convex polygon is always ```360 degrees``` -> + - -> ## Properties of Quadrilaterals ->> Determine the shape using the properties of it +## 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
```Perimeter```: 2(l+w)|| ->> |Triangle|```Area```: bh/2
```Perimeter```: a+b+c|| ->> |Circle|```Area```: πr2
```Circumference```: 2πr or πd|| ->> |Trapezoid|```Area```: (a+b)h/2
```Perimeter```: a+b+c+d|| - -> ## 3D Geometry Equations ->> |3D Object|Formula|Picture| ->> |:----|:------|:------| ->> |Rectangular Prism|```Volume```: lwh
```SA```: 2(lw+lh+wh)|| ->> |Square Based Pyramid|```Volume```: 1⁄3b2h
```SA```: 2bs+b2|| ->> |Sphere|```Volume```: 4⁄3πr3
```SA```: 4πr2|| ->> |Cone|```Volume```: 1⁄3πr2h
```SA```: πrs+πr2|| ->> |Cylinder|```Volume```: πr2h
```SA```: 2πr2+2πh|| ->> |Triangular Prism|```Volume```: ah+bh+ch+bl
```SA```: 1⁄2blh|| - - -> ## 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
A = lw
Amax = (w)(w)
Amax = w2|A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width.
P = 2(l+w)
Pmin = 2(w)(w)
Pmin = 2(2w)
Pmin = 4w| ->> |3-sided rectangle|l = 2w
A = lw
Amax = 2w(w)
Amax = 2w2|l = 2w
P = l+w2
Pmin = 2w+2w
Pmin = 4w| - - ->> ### 3D Objects - ->> |3D Object|Maximum Volumne|Minimum Surface Area| ->> |:--------|:--------------|:-------------------| ->> |Cylinder(closed-top)|The cylinder must be similar to a cube where h = 2r
V = πr2h
Vmax = πr2(2r)
Vmax = 2πr3|The cylinder must be similar to a cube where h = 2r
SA = 2πr2+2πrh
SAmin = 2πr2+2πr(2r)
SAmin = 2πr2+4πr2
SAmin = 6πr2| ->> |Rectangular Prism(closed-top)|The prism must be a cube,
where l = w = h
V = lwh
Vmax = (w)(w)(w)
Vmax = w3|The prism must be a cube,
where l = w = h
SA = 2lh+2lw+2wh
SAmin = 2w2+2w2+2w2
SAmin = 6w2| ->> |Cylinder(open-top)|h = r
V = πr2h
Vmax = πr2(r)
Vmax = πr3|h = r
SA = πr2+2πrh
SAmin = πr2+2πr(r)
SAmin = πr2+2πr2
SAmin = 3πr2| ->> |Square-Based Rectangular Prism(open-top)|h = w/2
V = lwh
Vmax = (w)(w)(w⁄2)
Vmax = w3⁄2|h = w/2
SA = w2+4wh
SAmin = w2+4w(w⁄2)
SAmin = w2+2w2
SAmin = 3w2| - -> ## 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 - ->> - -> ## Median ->> Each median divides the triangle into 2 smaller triangles of equal area ->> The centroid is exactly the way of each median from the vertex, or 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 - ->> - -> ## 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 ->>> ->> ```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 ->>> - -> ## 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``` ->> - -> ## 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 ```the rate of change``` ->> ```Slope Formula```: **m = (y2-y1)/(x2-x1)** ->> ```Standard Form```: **ax + by + c = 0**, a∈Z, b∈Z, c∈Z (must be integers and ```a``` must be positive) ->> ```Y-intercept Form```: **y = mx + b** ->> ```Point-slope Form```: **y2-y1 = m(x2-x1)** ->> The slope of a vertical lines is undefined ->> The slope 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 Lines ->> To find the perpendicular slope, you will need to find the slope point ->> Formula: slope1 × slope2 = -1 ->> Notation: m⊥ ->> - - -> ## 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 ->> - ->> **Direction is always referring to:** - ->> 1. ```go towards home``` ->> 2. ```going away from home``` ->> 3. ```stop``` - -> ## 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. ->> - ->> ### How To Determine the Equation Of a Line of Best Fit ->> 1. Find two points **```ON```** the ```line of best fit``` ->> 2. Determine the ```slope``` using the two points ->> 3. Use ```point-slope form``` to find the equation of the ```line of best fit``` - -> ## 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|.....| ->>|0|-1|(-1)-(-2) = 1| ->>|1|0|0 - (-1) = 1| ->>|2|1|1 - 0 = 1| ->>|3|2|2 - 1 = 1| ->>|4|3|3 - 2 = 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 ->>> - -> ## 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 = ax2 + bx + c** ->> D = b2 - 4ac ->> D > 0 ```(2 distinct real solutions)``` ->> D = 0 ```(1 real solution)``` ->> D < 0 ```(no real solutions)``` - ->> - -> ## 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 - ->> - ->> There are 3 possible cases ->> In addition, to determine the number of solutions, you the Discriminant formula **D = b2 - 4ac** - - -> # 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 ≥ mx + b| ->> |Shade the region ```below``` the line|y < mx + b| y ≤ mx + b| - ->> ## If - ->> |x > a
x ≥ a| ->> |:------------------| ->> shade the region on the **right** - ->> ## If - ->> |x < a
x ≤ 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 - ->> - - -> ## Tips ->> Read the questions carefully and model the system of equations correctly ->> Be sure to name your equations ->> Label your lines + |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
```Perimeter```: 2(l+w)|| + |Triangle|```Area```: bh/2
```Perimeter```: a+b+c|| + |Circle|```Area```: πr2
```Circumference```: 2πr or πd|| + |Trapezoid|```Area```: (a+b)h/2
```Perimeter```: a+b+c+d|| + +## 3D Geometry Equations +|3D Object|Formula|Picture| + |:----|:------|:------| + |Rectangular Prism|```Volume```: lwh
```SA```: 2(lw+lh+wh)|| + |Square Based Pyramid|```Volume```: 1⁄3b2h
```SA```: 2bs+b2|| + |Sphere|```Volume```: 4⁄3πr3
```SA```: 4πr2|| + |Cone|```Volume```: 1⁄3πr2h
```SA```: πrs+πr2|| + |Cylinder|```Volume```: πr2h
```SA```: 2πr2+2πh|| + |Triangular Prism|```Volume```: ah+bh+ch+bl
```SA```: 1⁄2blh|| + + +## 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
A = lw
Amax = (w)(w)
Amax = w2|A rectangle must be a square to minimaze the perimeter for a given area. The length is equal to the width.
P = 2(l+w)
Pmin = 2(w)(w)
Pmin = 2(2w)
Pmin = 4w| + |3-sided rectangle|l = 2w
A = lw
Amax = 2w(w)
Amax = 2w2|l = 2w
P = l+w2
Pmin = 2w+2w
Pmin = 4w| + + +### 3D Objects + + |3D Object|Maximum Volumne|Minimum Surface Area| + |:--------|:--------------|:-------------------| + |Cylinder(closed-top)|The cylinder must be similar to a cube where h = 2r
V = πr2h
Vmax = πr2(2r)
Vmax = 2πr3|The cylinder must be similar to a cube where h = 2r
SA = 2πr2+2πrh
SAmin = 2πr2+2πr(2r)
SAmin = 2πr2+4πr2
SAmin = 6πr2| + |Rectangular Prism(closed-top)|The prism must be a cube,
where l = w = h
V = lwh
Vmax = (w)(w)(w)
Vmax = w3|The prism must be a cube,
where l = w = h
SA = 2lh+2lw+2wh
SAmin = 2w2+2w2+2w2
SAmin = 6w2| + |Cylinder(open-top)|h = r
V = πr2h
Vmax = πr2(r)
Vmax = πr3|h = r
SA = πr2+2πrh
SAmin = πr2+2πr(r)
SAmin = πr2+2πr2
SAmin = 3πr2| + |Square-Based Rectangular Prism(open-top)|h = w/2
V = lwh
Vmax = (w)(w)(w⁄2)
Vmax = w3⁄2|h = w/2
SA = w2+4wh
SAmin = w2+4w(w⁄2)
SAmin = w2+2w2
SAmin = 3w2| + +## 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 + +- + +## Median +- Each median divides the triangle into 2 smaller triangles of equal area +- The centroid is exactly the way of each median from the vertex, or 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 + +- + +## 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 + - +- ```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 + - + +## 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``` +- + +## 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 + + +# Unit 5: Analytical Geometry and Linear Relations +- ```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 ```the rate of change``` +- ```Slope Formula```: **m = (y2-y1)/(x2-x1)** +- ```Standard Form```: **ax + by + c = 0**, a∈Z, b∈Z, c∈Z (must be integers and ```a``` must be positive) +- ```Y-intercept Form```: **y = mx + b** +- ```Point-slope Form```: **y2-y1 = m(x2-x1)** +- The slope of a vertical lines is undefined +- The slope 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 Lines +- To find the perpendicular slope, you will need to find the slope point +- Formula: slope1 × slope2 = -1 +- Notation: m⊥ +- + + +## 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 +- + +**Direction is always referring to:** + + 1. ```go towards home``` + 2. ```going away from home``` + 3. ```stop``` + +## 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. +- + +### How To Determine the Equation Of a Line of Best Fit + 1. Find two points **```ON```** the ```line of best fit``` + 2. Determine the ```slope``` using the two points + 3. Use ```point-slope form``` to find the equation of the ```line of best fit``` + +## 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|.....| + |0|-1|(-1)-(-2) = 1| +|1|0|0 - (-1) = 1| +|2|1|1 - 0 = 1| +|3|2|2 - 1 = 1| +|4|3|3 - 2 = 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 + +# Unit 6: System of Equations + +## 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 +- + +## 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 = ax2 + bx + c** +- D = b2 - 4ac +- D > 0 ```(2 distinct real solutions)``` +- D = 0 ```(1 real solution)``` +- D < 0 ```(no real solutions)``` + +- + +## 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 + +- + +- There are 3 possible cases +- In addition, to determine the number of solutions, you the Discriminant formula **D = b2 - 4ac** + + +# 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 ≥ mx + b| + |Shade the region ```below``` the line|y < mx + b| y ≤ mx + b| + +- ## If + + - |x > a
x ≥ a| + |:------------------| + - |shade the region on the **right**| + +- ## If + + - |x < a
x ≤ 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 + +- + + +## 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!!!!!!!!!!!** +- 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 ->> ```Lets``` Statement ->> ```Conclusion``` +## 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 +- ```Lets``` Statement +- ```Conclusion``` -> ## Graph Problems ->> Look up on tips in units (5) and (6) ->> be sure to use a ruler when graphing +## 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 +## 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 -> Ms Hung(Katie) - She helped me check over my study sheet, an amazing teacher! -> Magicalsoup - ME! +- Ryan Mark - He helped provide alot of information for me +- Ms Hung(Katie) - She helped me check over my study sheet, an amazing teacher! +- Magicalsoup - ME!