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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

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

Unit 1: Essential Skills

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

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)

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

Operations with Rationals

Q = { $\frac{a}{b}$ | 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
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:

$= \frac{3}{4} \div \frac{2}{14}$ [Reduce to lowest terms]
$= \frac{3}{4} \div \frac{1}{7}$ [Multiply by reciprocal]
$= \frac{3}{4} \times 7$
$= \frac{21}{4}$ [Leave as an improper fraction]

Shortcut for multiplying fractions

Exponent Laws

Rule	Description	Example
Product	a^m × aⁿ = a^n+m	2³ × 2² = 2⁵
Quotient	a^m ÷ aⁿ = a^n-m	3⁴ ÷ 3² = 3²
Power of a Power	(a^m)ⁿ = a^mn	(2³)² = 2⁶
Power of a Quotient	=	=
Zero as Exponents	a⁰ = 1	21⁰ = 1
Negative Exponents	a^-m =	1^-10 =
Rational Exponents	a^n/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

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.

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	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

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

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

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

Expand and simplify both sides
Isolate using reverse order of operations
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

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

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

Unit 4: Measurement and Geometry

Angle Theorems

Transversal Parallel Line Theorems (TPT)
1. Alternate Angles are Equal (Z-Pattern)
2. Corresponding Angles Equal (F-Pattern)
3. Interior Angles add up to 180 (C-Pattern)

Supplementary Angle Triangle (SAT)

When two angles add up to 180 degrees

Opposite Angle Theorem (OAT) (OAT)

Two lines intersect, two angles form opposite. They have equal measures

Complementary Angle Theorem (CAT)

The sum of two angles that add up to 90 degrees

Angle Sum of a Triangle Theorem (ASTT)

The sum of the three interior angles of any triangle is 180 degrees

Exterior Angle Theorem (EAT)

The measure of an exterior angle is equal to the sum of the measures of the opposite interior angles

Isosceles Triangle Theorem (ITT)

The base angles in any isosceles triangle are equal

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

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

Figure	Properties
Scalene Triangle	no sides equal
Isosceles Triangle	two sides equal
Equilateral Triangle	All sides equal
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 A_max = (w)(w) A_max = 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) P_min = 2(w)(w) P_min = 2(2w) P_min = 4w
3-sided rectangle	l = 2w A = lw A_max = 2w(w) A_max = 2w²	l = 2w P = l+w2 P_min = 2w+2w P_min = 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 V_max = πr²(2r) V_max = 2πr³	The cylinder must be similar to a cube where h = 2r SA = 2πr²+2πrh SA_min = 2πr²+2πr(2r) SA_min = 2πr²+4πr² SA_min = 6πr²
Rectangular Prism(closed-top)	The prism must be a cube, where l = w = h V = lwh V_max = (w)(w)(w) V_max = w³	The prism must be a cube, where l = w = h SA = 2lh+2lw+2wh SA_min = 2w²+2w²+2w² SA_min = 6w²
Cylinder(open-top)	h = r V = πr²h V_max = πr²(r) V_max = πr³	h = r SA = πr²+2πrh SA_min = πr²+2πr(r) SA_min = πr²+2πr² SA_min = 3πr²
Square-Based Rectangular Prism(open-top)	h = w/2 V = lwh V_max = (w)(w)(^w⁄₂) V_max = ^w³⁄₂	h = w/2 SA = w²+4wh SA_min = w²+4w(^w⁄₂) SA_min = w²+2w² SA_min = 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:

go towards home
going away from home
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

Find two points ON the line of best fit
Determine the slope using the two points
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

	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

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
Lets Statement
Conclusion

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
Ms Hung(Katie) - She helped me check over my study sheet, an amazing teacher!
Magicalsoup - ME!

35 KiB Raw Blame History Unescape Escape

Math Study Sheet!!!!

Exam Detail

Unit 1: Essential Skills

Simple Arithmetics

Addition / Subtraction

Multiplication / Division

BEDMAS / PEMDAS

Interval Notation

Pythgorean Theorem

Operations with Rationals

Example Simplify Fully:

Shortcut for multiplying fractions

Exponent Laws

Scientific Notation

Rates, Ratio and Percent

Number Lines

Tips

Unit 2: Polyomials

Introduction to Polynomials

Simplifying Alegebraic Expressions

Factoring

Shortcuts

Foil / Rainbow Method

Definitions

Tips

Unit 3: Solving Equations and Inequailties

Equations

Absolute Values

Quadractic Equations

Tips

Unit 4: Measurement and Geometry

Angle Theorems

Properties of Quadrilaterals

2D Geometry Equations

3D Geometry Equations

Optimization (For Maximimizing Area/Volume, or Minimizing Perimeter/Surface Area)

2D Objects

3D Objects

Labelling

Median

Terms:

Proportionality theorem:

Tips

Unit 5: Analytical Geometry and Linear Relations

Slope and Equation of Line

Relations

Perpendicular Lines

Definitions

Variables

Statistics

Time - Distance Graph

Scatterplot and Line of Best Fit

How To Determine the Equation Of a Line of Best Fit

Table of values

Tips

Unit 6: System of Equations

Linear System

Number of Solutions

Discriminant

Solving Linear-Quadratic Systems

Ways to solve Systems of Equations

3. Graphing

Solving Systems of Linear Inequalities

If

If

If

Tips

General Tips

Word Problems

Graph Problems

System of Equations

Credits

35 KiB

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