Kindergarten Maths 幼儿园数学入门

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Study notes mainly from Khan Academy. Not literally for Kindergarten maths only, but also including all K12 math subjects.

KHAN ACADEMY MASTERY rules collected:

• 1 skill means 1 practice subject, could be found at practice page on the subject.
• 1 skill includes 4 to 7 questions.
• 1 skill finished 75%, will promote to Practiced.
• 1 skill finished 100%, will promote to LEVEL ONE.
• 1 skill must be practiced to join the Mastery challenge.
• 1 question correct on Mastery challenge, the related skill will get 1 level up.
• 1 question incorrect on Mastery challenge, the related skill will get 1 level down.
• 1 quiz finished in course section, the correct skills will promote to LEVEL ONE.
• 1 unit test finished in course section, the correct skill will get 1 level up till MASTERY. Incorrect answer will get 1 level down.

Practice To-do List

• [x] Algebra basics
• [x] 7th grade
• [x] 8th grade
• [x] Algebra 1
• [x] Algebra 2
• [x] Geometry
• [x] Trigonometry
• [x] High school statistics
• [x] Precalculus
  • [x] Trigonometry
  • [x] Conic sections
  • [x] Vectors
  • [x] Matrices
  • [x] Complex numbers
  • [x] Probability and combinatorics
  • [x] Series

Khan Academy Mastery challenge

• [x] Early Math (100%)
• [x] Kindergarten (100%)
• [x] 1st grade (100%)
• [x] 2nd grade (100%)
• [x] 3th grade (100%)
• [x] 4th grade (100%)
• [x] 5th grade (100%)
• [x] 6th grade (100%
• [x] 7th grade (100%)
• [x] 8th grade (100%)
• [x] Arithmetic (100%)
• [x] Basic geometry (100%)
• [x] Pre-algebra (100%)
• [x] Algebra basics (100%)
• [x] Algebra 1 (100%)
• [x] Algebra 2 (100%)
• [x] Geometry (100%)
• [x] Trigonometry (100%)
• [x] High school statistics (100%)
• [x] Precalculus (100%)
• [x] Mathematics 1 (100%)
• [x] Mathematics 2 (100%)
• [x] Mathematics 3 (100%)
• [x] Linear Algebra (No mastery) (100%)
• [ ] AP calculus AB
• [ ] AP Statistics (No mastery)

Unit Tests (For reviewing forgettable High school level concepts)

Algebra (All contents):

• [ ] Linear equations, functions, & graphs (Slope, Linear equation forms)
• [ ] Sequences (arithmetic/geometric sequences)
• [ ] Functions (Domain, Range, Interval, ARC, Transformation, Inversion, Symmetry, End behavior, Periodicity)
• [ ] Quadratic equations & functions (Parabola, Quadratic forms, Transform quadratic functions)
• [ ] Polynomial expressions, equations, & functions (monomial, binomial, Zeros of polynomial func)
• [ ] Exponential & logarithmic functions (Exponent properties, Radicals, growth&decay, Properties of logarithms)
• [ ] Radical equations & functions (Extraneous solutions, Domain of radical functions)
• [ ] Rational expressions, equations, & functions (rational expressions, End behavior, Discontinuities)
• [ ] Trig functions (Radians, Graphs, Sinusoidal functions, Inverse, Trig identities)
• [ ] Complex numbers (Absolute value & angle, Distance & midpoint, Forms of complex numbers)
• [ ] Conic sections (Functions of conic shapes, Foci, Hyperbola, Ellipse)

Geometry:

Trigonometry:

Statistics:

Review hardcore quiz

• [ ] Trigonometry
  • [ ] Sinusoidal functions
  • [ ] Graph sinusoidal functions
  • [ ] Inverse trig functions
• [ ] Probability
  • [ ] Probability basics
  • [ ] Permutations and combinations
• [ ] Algebra 2
  • [ ] Functions
  • [ ] Quadratics & Parabola
  • [ ] Polynomials
  • [ ] Graphs of rational functions
  • [ ] Exponential and Logarithmic functions
  • [ ] Series
• [ ] Precalculus
  • [ ] Conversion of complex numbers

Table of Contents

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How To Read Math Notations

Collect some common maths notions.

Refer to wiki: List of mathematical symbols

For further information, review this link, or download this pdf: Guidelines for Reading Mathematics.pdf. Notice that: Word in PARENTHESIS doesn't have to say out.

Product

"Three times four" "Three multiply by four".

3 × 4

"Open parenthesis x plus three close parenthesis multiply by open parenthesis ...." "x plus three, multiply by x minus two."

(x+3)(x-2)

Quotient & Fraction

"Three fourth " "Three over four".

3/4

"One half" "One over two"

1/2

"Three halves" "Three over two".

3/2

"x plus one over x minus one" .

(x+1)/(x-1)

exponent

"x squared" "x (raised) to the second (power)"

x²

"x cubed" "x to the third (power)"

x³

"Three to the zeroth power"

3⁰

"Nine to the a plus b (power)."

9⁽ª⁺ᵇ⁾

"Five to the three t (power)."

5³ᵗ

Factorial !

"4 factorial" "4 shriek" "4 bang"

4!

composite function

"f of x."

f(x)

"g of f of x."

(g◦f)(x)

"h of g of minus six".

(f◦g)(-6)

Set

"x in B", "x belongs to B", or "x is an element of B"

x ∈ B

"y not in B", or "y does not belong to B"

y ∉ B

Subset

"A is a subset of B", or "A is contained in B"

A ⊇ B

"B is a superset of A", or "B includes A", or "B contains A"

B ⊇ A

The relationship between sets established by ⊆ is called inclusion or containment.

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How to use calculator

Latex syntax

These syntax works for Chrome address bar, Mac Spotlight bar, Cymath etc.

• Exponent: for 3⁴, type 3^4, results 81.
• Root: for √9, type 9^(1/2), results 3.
• Logarithm: for log₂8, type log_2(8), results 3.
• Logarithm base-e: for ln 6, type ln 6, results 1.791....
• Logarithm base-10: for log 10, type log 10, results 1.

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Triangle types

Refer to Khan academy: Triangle types

Types by side length

• Scalene triangles with all different sides （sounds "scale-lin")
• Isosceles triangles with 2 equal sides (sounds "i-saw-sillys")
• Equilateral triangles with 3 equal sides (sounds "e-qui-lateral")

Types by angles

• Acute triangles with 3 acute angles
• Obtuse triangles with 1 obtuse angle
• Right triangles with 1 right angle.

Angle types

Types of angles: acute, right, obtuse(sounds "ob-tuse"), and straight.

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Dividing numbers strategy

Khan page.

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Polyhedra - 3D shape

Polyhedra, or Polyhedrons is plural for Polyhedron. means a 3D shape with surfaces all flat. Examples: Cube, Triangular Prism, Pyramids, Tetrahedron

More about the list of polyhedrons: Animated Polyhedron Models

Refer to math is fun: Polyhedron

Refer to Khan academy lession.

Regular polyhedrons:

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Prisms (3D shape)

A Prism is a 3D shape, which could be stretch out from a 2D shape with sides all straight. The 2D shape is called Cross-Section. A prism is a polyhedron, which means all faces are flat!

Refer to math is fun: Prisms - 3D shape

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Mean, median, & mode numbers (STATS)

Khan lecture.

• Mean number is just an average of all numbers listed.
• Median number is the middle positioned number in a ordered number set (means no duplicates). If there're two middles, then average them to get a median number.
• Mode number is the number shows up most times in a list.

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Dependent variables & independent variables

Refer to math is fun: dependent variable Refer to math is fun: independent variable

Understand:

• Independent variables: Input value of a function. (usually x)
• Dependent variables: Output value of a function. (usually y)

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Box plots and Quartiles (STATS: Distribution graph)

It's also called Box and whisker plots, or Five-number summary.

Khan lecture 1. Khan lecture 2. Maths is for fun Wiki.

Five-number summary

Interquartile range (IQR) (STATS: Box plot)

Refer to Khan academy.

Example

Example

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Clusters, gaps, peaks & outliers (STATS: Distribution graph)

Khan lecture: Shape for distributions. Khan lecture 2 Clusters, gaps, peaks & outliers.

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What are sine, cosine, tangent? (TRIGONOMETRY)

• Khan short video
• Maths is Fun

First to know the terms

It's only applicable to an Right triangle. A right triangle is a triangle with a 90-degree angle. The angle we pick out, is called Θ, theta. In vector related problems, it's also called the direction angle, see this Khan practice.

The side opposite to that angle is called Opposite. The side next to the angle is called Adjacent, "A-Jason-t". The side is long is called Hypotenuse, "High-po-ten-news`.

How to calculate sine, cosine and tangent of an angle

REMEMBER SHORTCUT: SOH-CAH-TOA, pronounced "so-kah-tow-ah".

• Sin = Opposite / Hypotenuse
• Con = Adjacent / Hypotenuse
• Tan = Opposite / Adjacent

How to calculate the ratio of an angle?

We want to know what the ratio of an angle is, 60° or 59°? If we know tan(x) = 123, whatever, then we can get it by x = tan_inverse(123). tan_inverse(...) or tan⁻¹(...) are the same.

Reciprocal trig ratios: Cosecant, Secant, Cotangent

Khan notes.

• Cosecant: csc(θ) = 1 / sin(θ)
• Secant: sec(θ) = 1 / cos(θ)
• Cotangent: cot(θ) = 1 / tan(θ)

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Terms, factors, & coefficients in Algebra

Khan lecture.

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Mean absolute deviation (MAD) - STATS

The deviation is the distance from the value to the mean value. It's used to describe how the values looks like or how they're laid on the axis, are they close to each other or far away.

The Mean absolute deviation is the absolute average of all deviations.

Khan lectures.

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Common 2D shapes

Quadrilaterals

Polygons

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Distance between points & lines

Distance of two points on x-y axis?

Refer to Khan academy lecture: pythagorean-theorem-distance

▼ Just to apply the Pythagorean Theorem:

Example

What is the distance between (-6, 8) and (-3, 9) ? Solve:

• Easy way to do this:
• Make a triangle with two points same with the given points, and solve the length of the triangle.
• So to solve the example, just like this:

Distance between point & line

Refer to Khan academy: Distance between point & line

Slope of a perpendicular line is the Negative Inverse of the slope of the given line.

Strategy:

• Draw a perpendicular line form the point to the given line.
• Find out the linear equation for this perpendicular line by:
  • Find the slope: which is the Negative Inverse of the slope of the given line:
  • Find the shift: plug in the point's x & y coordinates and get the shift
• Set two lines' equations equal and get the Intersect point's coordinates.
• Calculate two points distance by Pythagorean Theorem:

Example

Solve:

• Find the perpendicular line from the point to the given line:
  • Slope = -1/-1 = 1
  • y=ax + b -> 2 = -3*1 +b -> b=5 -> y=x+5
• Find the intersect: -x+1 = x+5 -> x=-2 -> y=3 -> intersects at (-2, 3)
• Calculate distance between two points:
• The answer is √2

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Makeup price problem

A markup rate is a percentage of the wholesale price that a store adds to get a selling or retail price.

Question

Answer

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Pythagorean Theorem (TRIANGLE)

勾股定理。 Pronounced: Pe~'tha-gor-rean 'Theor-rem In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Refer to Math is fun: Pythagorean theorem

Triangle inequality theorem

Refer to Math is fun: Triangle inequality theorem

In a triangle, the sides always follow these two rules:

• Any side of a triangle is always shorter than the sum of the other two sides
• The third side must be also larger than the difference between the other two sides

Example:

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Complementary, supplementary, vertical angles

Refer to Khan academy: Complementary, supplementary, vertical angles

Complementary angles are two angles with a sum of 90°. A common case is when they form a right angle.

Supplementary angles are two angles with a sum of 180°. A common case is when they lie on the same side of a straight line.

Vertical angles are angles opposite each other where two lines cross. A pair of vertical angles have the same measure of angle.

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Transformations (GEOMETRY)

Transformation means something's changing, transforming.

Refer to Khan academy: Transformations

Types of transformations:

• Translations: Slide or move the shape. In geometry, a translation moves a thing up and down or left and right.
• Rotation: Turn or rotate the shape. In geometry, rotations make things turn in a cycle around a definite center point.
• Reflection: Flip or mirror the shape. A reflection is a transformation that acts like a mirror: It swaps all pairs of points that are on exactly opposite sides of the line of reflection.
• Dilation: Expand or shrink the shape.

And, you can group above types of transformations into two groups: Rigid transformations and Dilations

Rigid transformations and Congruent

Rigitd transformations means you play around the shape without expanding or shrinking it. Or say, without dilation, all translation/rotation/reflection would be rigid transformation.

Congruent is the shape after you "rigid transformed`. Or say, without dilation, after all translation/rotation/reflection, the shape is called "congruent" to the original shape.

Khan lecture.

Dilations

Dilation, just a fancy word for "resizing", or "scaling".

Notice that: Dilation DOES NOT change any angle within the shape. Don't get confused with a horizontal stretch, which does change both sides and angles.

Similarity

When you resize, or say dilate a shape, you call the new shape similar to the original one. If nothing changed with the shape, you call it congruent to the original one.

Scale factor

Means how much you scale the shape, like 2 times bigger, or 2/3 smaller.

Notice:

• The scale factor is of the length of the shape, NOT the area of it.

Scale factors and area

When you scale the shape, the area of the new shape is (scale facto)² times to the original one.

Khan lecture: Scale factors and area

For Shape A and scaled shape A', it leads to two practical conclusions:

• If we know the scale factor is x, then the area of A is x² times to the original one.
• If we know the ratio (area of A') ÷ (area of A) is x, then the scale factor is √x

Example

Solve:

• New area G is 1/9 of F
• So the (scale factor)² = 1/9, which results the scale factor = 1/3

Dilation Center

"Dilate the shape ABOUT a point P", means take the point as a center to dilate the shape.

How does it work? As the picture below, just simply scale the distance from each vertex(point) of the shape to the point.

How to find the dilation center?

The point P and it's image and dilation center, should be IN ONE LINE !

In the example below, you should forget about the origin but set the P point as origin and count the distance of each vertex of the triangle:

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Area, perimeter, surface, volume of shapes (Geometry)

Cheatsheets for calculating area, perimeter, surface, volume of common shapes.

Area of plane shapes

Triangle, square, rectangle, parallelogram, trapezoid, circle, ellipse, sector.

Refer to math is fun: area

Areas:

• Area of Equilateral Triangles:
• Isosceles Right Triangle:

Surface Area

• Surface of Cylinder: 2π · r · (r + h)
• Surface of Sphere: 4π · r²
• Surface of Cone: π · r( r + √(h²+r²) )

Volume of Cuboids, Rectangular Prisms and Cubes

Volume = Length · Width · Height

Volume of Cone, sphere, Cylinder

The volume of a Cylinder is: π · r² · h The volume of a Cone is: 1/3 π · r² · h The volume of the Sphere is:4/3 π · r³

Refer to math is fun: cone-sphere-cylinder

Volume, surface area of Pyramid

Refer to math is fun: pyramids

• The Volume of a Pyramid: 1/3 · [Base Area] · Height
• The Surface Area of a Pyramid:
  • When all side faces are the same: [Base Area] + 1/2 Perimeter · [Slant Length]
  • When side faces are different: [Base Area] + [Lateral Area]

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Rational & irrational numbers

A rational number can always be expressed as a fraction of two integers. vice versa, a irrational number cannot be expressed as a fraction of two integers.

Refer to math is fun: Rational & irrational numbers

A square root of a non-perfect square is an irrational number, because it cannot be expressed as the fraction of two integers.

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Stem and Leaf Plots (Stats)

A Stem and Leaf Plot is a special table where each data value is split into a "stem" (the first digit or digits) and a "leaf" (usually the last digit).

Refer to khan academy: Stem and Leaf Plots Khan lecture.

In the above plot:

• The stem column means the tens number of the whole number.
• The row column means the ones number of the whole number. For example, the numbers of the second row represent: 12, 13, 15.

How the Stem and Leaf Plot represent numbers, it depends on the context. Stem can represents number of tens place, or ones place; Leaf can represents ones place or even decimal places.

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Proportional relationship

Linearly, if x and y has a relationship that one is another's proportion, then they have a proportional relationship. Expression as: y/x = k.

Or it's a version of linear equation y=mx +b, only the b=0, then y=mx.

A relationship is proportional if its graph is a straight line through the origin. Remember that the origin is the point (0,0)

At the showing image above, only first one shows a proportional relationship. The other two are not linear and going through origin.

Directly Proportional & Inversely Proportional

Refer to Maths is fun.

• Directly proportional: As one amount increases, another amount increases at the same rate.
• Inversely Proportional: when one value decreases at the same rate that the other increases.

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Three Terms ratio problem

It takes 54 minutes for 4 people to paint 6 walls.
How many minutes does it take 6 people to paint 7 walls?

• 4 p 1 wall: 54 / 6 = 9 minutes, then:
• 1 p 1 wall: 9 * 4 = 36 minutes, then:
• 1 p 7 wall: 36 * 7 = 252 minutes, then:
• 6 p 7 wall: 252 / 6 = 42 minutes.

Note that, it cannot be easily get out the result, but only to understand the context and make it out step by step.

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Scientific Notation

Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers: It makes it easy to use big and small values.

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The Slope of a line

Refer to math is fun: slope

• Rise: the vertical change is called "rise".
• Run: the horizontal change is called "run".

• Positive slope: the line is going up.
• Negative slope: the line is going down.
• Slope of zero: a horizontal line.
• Undefined slope: a vertical line.

Slope-intercept form

y = mx +b is called "slope-intercept form` of an equation.

• m is the slope of this line
• b is the y-intercept of this line.

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Polynomials

Refer to math is fun: Polynomials

Just an expression of some values, including variables, exponents, constants... It only fits to "poly" "nomials", aka. multiple terms, as below:

Degrees of a polynomial

Means the highest exponent of a variable in the polynomial.

For example, if it's a quadratic, then it's a 2nd degree polynomial.

Monomial, Binomial, Trinomial

Multiply binomials

Refer to math is fun: special-binomial-products

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Quadratics

Refer to math is fun: Quadratics

Different forms a quadratic

There're many tricks to convert one form to another, mainly depends on what information we have and what information we'd like to get.

Common forms for quadratic as below:

• Standard form:
• Two binomials form: Could get two solutions without calculator.
• Perfect square form: Could get the positions of vertex by only eyeballing it. In another way, you could tell how much the graph was moved from the origin point.

What is the Solutions to the quadratic

The solutions to the Quadratic Equation are where it is equal to ZERO.

They are also called ROOTS, or sometimes ZEROS, or REAL ZEROS, OR REAL NUMBER ZEROS, or DISTINCT REAL NUMBER ZEROS. Graphically, when the graph touches x-axis, the point is A SOLUTION to the quadratic equation.

There're different ways to get solutions for the quadratic:

• Factor the quadratic as a product of two binomials
• Re-group the quadratic, and then factor it
• Use Quadratic Formula.

To factor a quadratic as the product of two binomials

To find out the answer quickly and get intuition of it step by step, visit this cheat online solver: Cymath.

It's easy to expand product of two binomials to a quadratic, but tricky to factor them back.

What we should do is, in the standard quadratic form ax² + bx +c = 0, we are to:

• Find the two common factors of "c"
• Be sure the sum of two factors equals to "b" As the example below, factors of "c" could be 4 and 6, and the sum of them is 10. So the answer is (x+4)(x+6)

To factor quadratics by re-grouping (to factor more complicated quadratics)

Refer to khan academy: factoring-by-grouping

When the coefficient of the 2nd degree term is not 1, things get more complicated.

Khan notes 1. Khan notes 2.

The way to do it is, in the standard form of Ax²+Bx+C:

• Find two numbers add up equal B
• The product of the two numbers should equal to A×C
• Regrouping Bx to two terms of x with the two number.

Use Quadratic formula to solve equation

Refer to math is fun: quadratic-equation-derivation

If we can't easily factor the quadratic, we have to use the ultimate formula:

It's an universal method for solving any quadratic equation.

Discriminant of a quadratic

Refer to khan academy: discriminant-review

Within the quadratic formula, there is a b2 − 4ac called Discriminant.

Refers to Mathwarehouse.

The solutions table as below:

The meaning of it, is to tell us about the solution of a quadratic:

• when b2 − 4ac is POSITIVE, we get TWO Real solutions. (It touches x-axis twice.)

• when it is ZERO we get just ONE Real solution. (Its vertex is on x-axis.)

• when it is NEGETIVE we get TWO Complex solutions. (It won't touch x-axis at all.)

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Parabola

Refer to math is fun: Parabola : The graph for quadratic

Parabola is a quadratic's graph.

A parabola always has a vertex, either its top point or bottom point. If you draw a vertical line goes through the vertex, it can divided the graph into two mirrored part.

The Concavity of parabola

• When it opens up, we also call it Concave up,
• When it opens down, we also call it Concave down.

Equation forms of a Parabola

Refer to math warehouse: Equation forms of a Parabola

There're two common forms of equation to express the parabola: Standard form and Vertex form.

And there's another form for the parabola: the Factor form.

How to get the Axis of symmetry

Refer to math warehouse: How to get the Axis of symmetry

For a parabola, the Axis of symmetry goes through the vertex, which represent the x position of the vertex.

The formula equation of Axis of symmetry:

• In the Vertex form of quadratic y = (x-h)² + k, the symmetry line is: x = h. Mind that, to understand this form, refers to the Absolute value graph notion.

• In the Standard form of quadratic y=ax²+bx+c, the symmetry line is: x=−b/2a

• In the Factor form of quadratic y=c(x-a)(x-b), the symmetry line is: x=(a+b)/2

How to get the Vertex

Getting the Axis of symmetry is half way to get the vertex, since it can only represent the x position of vertex. For getting the y position, just input the x value into the equation, and get the y value. Solved.

It's easier to get the vertex in the factor form of quadratic:

Then the vertex is (h, k).

How to graph the quadratic

To graph a quadratic, we have a few different ways, and each needs the information of a few points:

• A vertex point, two root points.
• A vertex point, a y-intersect point.
• A vertex point, any two points on the graph.

Refer this page to review how to get a parabola's intersects.

• To get two roots: Just solve the equation and get two solutons (if there're two solutions).

• To get the y-intersect point: Simply let the x=0, and solve the quadratic to get y position of the vertex. And input the y value to the equation, to solve x position. Then we get the vertex: (x, y)

• To get any two points of the graph: Just ASSUME two x positions, then input to the quadratic, to get y positions. Then we get: (x1, y1) and (x2, y2)

Parabola from geometric perspective

Refer to khan academy: Parabola from geometric perspective

Definition: A parabola is the set of ALL POINTS whose distance from a certain POINT (the focus) is equal to their distance from a certain LINE (the directrix).

It means,

To draw a parabola, you only need a point and a line.

Khan lecture. Maths if fun.

Equation for the parabola

In this graph above, we will get an equation by Pythagorean Theorem

And expand that equation, we got a parabola equation in Vertex quadratic form:

y = c (x - i)² + j

So we could spot out the focus and directrix information from a vertex quadratic equation. In which, the (i, j) represent the vertex, (h, k) represents the focus, and the y=k represents the directrix.

Equation of a parabola from focus & directrix

[Khan practice.](Equation of a parabola from focus & directrix)

Example

Solve: According to the focus and directrix information, we know we can consist an equation like:

(y-k)² = (x-a)² + (y-b)²

But that's for a vertical parabola. When the directrix is obviously making it to a horizontal parabola, we have to change the equation to:

(x-k)² = (x-a)² + (y-b)²

So it becomes:

(x+7)² = (x+3)² + (y+5)²

Expand the equation to get:

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Bisect

You can bisect a line segment, an angle, a shape, etc.

Bisector is the point or line or anything you use to divide things. An angle bisector is a line, or you could say you draw a line bisects the angle.

Refer to math is fun: Bisect Bisect is a verb, means to divide into two equal parts.

Angle Bisector Theorem

Refer to math bits notebook: Angle Bisector Theorem When we draw an angle bisector in a triangle, we get two equal ratio of related lines.

It could be also two ratios like this:

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Cross Sections (GEOMETRY)

Refer to math is fun: Cross Sections (Geometry)

Cross sections is a interesting geometry problem, it let you thinks about how to slice a 3D solid shape into some 2D shapes, for instance:

• Slice a cube into:
  • triangle (3 sides)
  • square (4 sides)
  • rectangle (4 sides)
  • pentagon (5 sides)
  • hexagon (6 sides)
• Slice a pyramid into:
  • triangle (3 sides)
  • square (4 sides)
  • rectangle (4 sides)
  • trapezoid (4 sides)
  • pentagon (5 sides)
• Slice a triangular prism into:
  • triangle (3 sides)
  • square (4 sides)
  • rectangle (4 sides)
  • pentagon (5 sides)
• Slice a cylinder into:
  • circle (no straight sides)
• Slice a cone into:
  • triangle (3 sides)
  • circle (no straight sides)

Note:

• To slice it must be using a plane to cut through.
• You can not only slice the 3D shape horizontally and vertically, but also any angle you like to get it.

Slice a cube to get a pentagon:

Slice a triangular prism to get a pentagon:

Slice a right pyramid to get a pentagon:

Slice a cone to get Ellipse:

Slice a cone to get a parabola:

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Intro to Probability

It's easy but always confusing if you haven't yet totally understood it in the first place.

The very first thing to do for solving a probability problem, is to CATEGORISE the problem and apply different formula.

• One event
• One event repeats
• Independent events in sequence

ONE event

• The probability of an event can only be 0 to 1 (or 0% to 100%).
• The probability of event A is often written as P(A).
• The probability of an event with condition is often written as P(condition).

Common cases:

• Flip a coin: P(head) = 1/2.
• Roll a die: P(>3) = 3/6

Theoretical & experimental probabilities

The formula Fav outcomes / Total outcomes only gives you the Theoretical probability. But when you do some experiments, like flip a coin 10,000 times, and you may find out the probability of the result of experiments is way so different than the theoretical one.

One event repeats

It's asking for

Example: Roll a die 100 times, how many times will you get a number greater than 3? Answer: P(>3) = 3/6 *100 The probability is 50 times.

Multiple events

Independent events in sequence

A & B happening

Independency

To understand probability, we really need to differentiate independent events and dependent events.

Khan lecture: Compound probability of independent events.

Coin flips are INDEPENDENT events: What happens in the first flip in no way affects what happens in the second flip.

And this is actually one thing that many people don't realise.

Gambler's Fallacy

There's someone who thinks, if he got a bunch of heads in a row, then all of a sudden, it becomes more likely on the next flip to get a tails.

THAT IS NOT THE CASE.

Every flip is an independent event. What happened in the past in these flips does not affect the probabilities going forward.

Sample space

A dummy method, just to draw a table or a tree shows every outcome it could be, and pick out all favourable results.

Example:

First to notice that, it's ONE event.

Example:

Example:

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Convert Repeating decimal to fraction

Refer to khan academy: Convert Repeating decimal to fraction

There're some repeating decimals or recurring decimals, it's way more easier to write down with a fraction.

Khan lecture 1. Khan lecture 2.

An online solver here: Calcul.com

The trick to do so is, to set up two numbers multiplied by 10 or 100 or 1000 or greater, letting their repeating part to fit to each decimal places. Like this:

or this,

or even this,

Here're some examples to convert repeating decimals to fraction:

• For simple repeating: set the number as x, then for example:

• For more decimals repeating:

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The Evolution of Numbers

refer to maths is fun.

Classification of numbers

Refer to Khan academy.

It's easy to classify most of the times, just to notice that:

• Whole number means positive integers and zero.
• Rational number must be able to represent as a fraction of two numbers.

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Equivalent systems of equations

The word equivalent here for system of equation, means both systems have the same solution. Equations in two systems, don't have to be the same thing at all. If you draw a line for each equation, there could be very different lines for each one.

Let's say it's in 2D, that term here literarily means two systems of 4 lines end up crossing at the same point (x,y), aka the solution.

Refer to khan academy: Equivalent systems of equations

How to determine if two systems are equivalent

You could either solve the solution for each system then compare the solution, or just to compare each equation with the counterpart(only in a easy case).

Solving method: Elimination

Ordinary way at middle school level: to eliminate variables and solve each variable one by one.

Khan review.

Solving method: Compare equations

Khan example.

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Solutions for systems of equation

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Proportional relationships

Refers to a function like y = mx. y and x have a proportional relationship, means they must go through a point (0,0).

Unit rate of proportional relationship

The unit rate of change of y with respect to x is the amount y changes for a change of one unit in x.

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Compound inequalities

Refer to khan academy: Compound inequalities

Means an inequality that combines two simple inequalities, and our mission is to find out the solution of the compound inequality.

There are two kind of combination with relationship AND and OR:

Solve compound inequalities

Just solve each simple inequality get the range of x, then compare two ranges in graph or in mind to get conclusion. Mind that, if it's in AND case, you need to take the overlap of two ranges; If it's in OR case, you have to take both two ranges as solution.

Example for OR: we get x ≥ 2 and x > 1 Since it has overlap and it's in OR case, so we take all ranges, as x > 1 can represent all.

Example for no solution: To solve this and get: x > -1 and x < -1 You can't be greater and less than something at same time, so no solution for this case.

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Dimensional analysis

Means to convert a unit to another unit.

Refer to Wiki: Dimensional analysis

For example, convert 60 km/h to 60000 m/h, or to 1 km/minute. It works out many problem in physical science and engineering problems.

Another example, to convert 10 mile/hour to meter/second:

There're some tricks to convert units: Khan lecture: Intro to dimensional analysis Math is fun: How to Safely Convert From One Unit to Another

Another example:

We need to convert both L and min, so for L we can multiply it by 1000 mL/L just to remove L. Because 1000 mL/L equals to 1, it's safe to do so. Similarly, for 1/min we can multiply it by min/60sec to remove min. To see below:

to simplify this equation we will get:

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Point-slope form of linear equation

Refer to khan academy: Point-slope form of linear equation

We all know the slope-intersect form of a linear equation is y = mx + c, which m is the slope of line, c is the constant represent the y-intersect of the line. And we can convert it to another form: which contains m for the slope of line, and point (a,b) for the point the line goes through.

Sometimes we only know the slope and an information of a point, so that can make us an linear equation too, simple like that.

Example: Q: Write the point-slope equation of the line that passes through (7,3) whose slope is 2. Solve: As for the point-slope form of linear equation y - b = m(x - a), we can rewrite it to:

y - 3 = 2(x - 7)

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Forms of linear equation

• Slope-intersect form: y = mx + b, which m as slope, b as y-intersect. This form is good for graphing a line with y-intersect and a certain slope.

• Slope-point form: y - b = m(x - a), which m as slope, (a, b) as a point on the line. This form is good for draw a line through one or two points.

• Standard form: Ax + By = C, which A,B,C are all integers. This form is good for graphing a line through both x and y intersects.

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Maxima and Minima of Functions

Refer to math is fun: Maxima and Minima of Functions

• Absolute maximum: means the greatest value of the function f(x).
• Absoute minimum: means the least value of function f(x).
• Relative maximum or called Local maximum: means the top of a hill on the function's graph.
• Relative minimum or called Local minimum: means the bottom of a valley on the function's graph.

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ARC: Average rate of change

Mind that: it's very basic idea for Differential calculus.

• For non-linear function, it means the Average slope for a specified domain. Because the slope is almost always changing in a non-linear function, and we have to specify a range or a domain then to get the average slope of this part of function.
• For a linear function, the Average rate of change just means the only one slope, which is constant.

For example:

We specify a domain on x-axis that from 0 to 3, aka. [0,3]. To know the average rate of change or the average slope, we can just directly draw a line connecting two points, and get the slope of this line.

Quiz 1:

Solve:

Quiz 2:

First to get the x's interval for [-3,-1]. Then we are to apply this equation to get ARC:

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Arithmetic Sequences

Refer to khan academy: Arithmetic Sequences

In arithmetic sequences, there's a common difference, which means the difference between neighbour terms is always the same.

We can represent the sequence as a formula. In the formula, we set the 1st term as k, and common difference as d. There're two forms of formula to represent the sequence:

• Explicit form:

• Recursive form:

Converting recursive & explicit forms of arithmetic sequences

Refer to Khan academy: Converting recursive & explicit forms of arithmetic sequences

Geometric sequences

Refer to khan academy: Geometric sequences

Similar idea for arithmetic sequences, only it's multiplying a common ratio here for geometric sequences, instead of adding a common difference in arithmetic sequences.

The common ratio below is 2:

There're also two forms for geometric sequences' formula:

• Explicit form:

• Recursive form:

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Absolute value graphs (Transformation)

Refer to Khan academy: Absolute value graphs

To understand this idea, need to review the previous knowledge of Transformation of graphs, including translation, rotation, reflection, dilation. Also need to know the scale factor of dilation.

For a simplest form of an absolute value equation, it can be represented as: And the most common use for this idea, is in Quadratic: y=x²

And the general form of an absolute value equation: In the equation,

• The sign of a, negative sign of a means the graph is flipped by x-axis.
• a is the scale factor of the graph, equals to g'(x)÷g(x).
• h is distance of moving right, should be subtracted by x.
• k is distance of moving up, should be added by y.

Notice that: The h is confusing sometime ---- it's the distance of moving right, and should be subtracted from x. Subtracted ,subtracted, subtracted! For understanding why is it subtracted, review this Khan lecture in 2 minutes.

Simplest equation y = |x|

Shifted(Translated) equation y = |x + h| + k

Flipped(Reflected) version y = - |x|

It's only flipping by the x axis.

Scaled(Dilated) equation y = a |x|

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Linear growth vs. Exponential growth

Refer to khan academy: Linear growth vs. Exponential growth

Quite a bit like the Arithmetic sequences vs. Geometric sequences,

except in Exponential relationship, it's using a common exponent, instead of using a common ratio in the geometric sequences.

~for which linear relationship and exponential relationship refers to the combination of x-sequence and y-sequence.~

The term relationship is applying to the value of f(x), aka. the y. If each value of y could have a constant gap, then the function f(x) has a linear relationship. If each value of y could have a constant ratio or constant exponent, then the function f(x) has a exponential relationship.

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Midpoint of a Line Segment

Refer to math is fun: Midpoint of a Line Segment The midpoint is halfway between the two end points of the line segment.

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Polygons (2D shapes)

Refer to math is fun: Polygons (2D shapes)

A Polygon is any flat shape with straight sides.

Interior Angles of Polygons

Refer to math is fun: Interior Angles of Polygons

Each time we add a side (triangle to quadrilateral, quadrilateral to pentagon, etc), we add another 180° to the total

So the sum of interior angles of a 2D shape is shown below:

• Any Polygon: (n-2)×180°
• Ray: 0°
• Triangle: 180°
• Quadrilateral: 360°
• Pentagon: 540°
• Hexagon: 720°
• Septagon: 900°
• Octagon: 1080°
• Nonagon: 1260°

Regular Polygons

In a regular polygon, which means all interior angles have same measure, each angle equals to sum ÷ n.

For example, in a 3-sides triangle, if it's a regular triangle, aka. equilateral triangle, each angle should be 180°/3, which is 60°.

Exterior angles of Polygons

The Exterior Angles of a Polygon add up to 360°, regardless how many sides it has.

Area of Irregular Polygons

Refer to math is fun: Area of Irregular Polygons

The first step is to turn each vertex (corner) into a coordinate,

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Function Transformations

Refer to math is fun: Function Transformations

Express different transformations in the function.

Refer to the previous note Graph transformations.

Let's assume:

• the function is f(x) = x^2 (it could be any other function instead).
• and the transformed new function is g(x).

Translation (Shift)

g(x) = f(x) + C

• If C is positive, the graph of f(x) moves UP.
• If C is negative, the graph of f(x) moves DOWN.

g(x) = f(x+C)

• If C is positive, the graph of f(x) moves LEFT.
• If C is negative, the graph of f(x) moves RIGHT.

Reflection (Flip/Mirror)

• Reflect the graph about the x-axis (Mirror it upside down, or downside up).

  g(x) = -f(x)

• Reflect the graph about the y-axis (Mirror the graph left to right, or right to left)

  g(x) = f(-x)

Dilation (Scale/Stretch/Compress)

Stretching the graph when C bigger than 1, compress it when C less than 1.

• Scale the graph about the y-axis (Vertically stretch or compress)

  g(x) = C・f(x)

• Scale the graph about the x-axis (Horizontally stretch or compress).

  g(x) = f(C・x)

Summary:

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Inverse functions

Refer to math is fun: Inverse functions

If a function means Map to, then inverse function means Map back. f(x)=y maps x to y, then f'(y)=x maps y back to x. Note that f' is the inverse function of f.

Khan notes.

When we have a function, it means we have a mapping rule. In this rule, we can map x to y, expressed as f(x) = y. But if we know y, how can we map back to x?

Identify inverse functions by composition

Refer to khan academy: Identify inverse functions by composition

To verify two functions are inverse, we can compose them.

If f(g(x)) = g(f(x)), then they are inverse.

invertible functions

Refer to khan academy: invertible functions

Not all functions have inverses. Those who do are called "invertible."

If a function f(x) maps x to y, and another function f'(y) can help us mapping y back to x, then the function f(x) is INVERTIBLE. If not, then it is NOT invertible.

Horizontal line test

On the graph of function f(x), if we can draw a horizontal line and ends up touches the graph more than once, we could say the function is NOT INVERTIBLE.

Inverse trig function word problems

EXAMPLE

Solve:

• Set equation Y(t) = −80
• Simplify equation and solve for all possible solutions:
  • base solution as θ = 1.7911 + 2πn
  • Use trig identity cos(θ) = cos(2π - θ) to get second solution as θ = 4.485 + 2πn.
• Replace θ with the expression of t and solve for t:
  • t*2π/3 = 1.7911 + 2πn , and solve it to get t = 0.9 + 3n
  • t*2π/3= 4.485 + 2πn, and solve it to get t = 2.1 + 3n
• Study the graph of trig function (review the Sinusoidal Functions), get informations:
  • Max at (0, -31)
  • Min at (1.5, -111)
  • Period is 2π/(2π/3) and get the period is 3
• According to the situation we know:
  • It passed the 1st Min and is before the 2nd Max
  • So it's over 1.5 sec at Bottom and not yet 3 sec at top.
• So within the range (1.5, 3), filter out invalid solution, 2.1 is the answer.

EXAMPLE

Solve:

• Simplify and solve to get solutions:
  • θ = 0.423 + 2πn
  • θ = 2.718 + 2πn, (with trig identity sin(θ) = sin(π-θ) when θ is positive)
• Replace θ with expression of t:
  • t = 24.57 + 365n
  • t = 157 + 365n
• Study the graph of trig function and get informations:
  • Period: 365
  • Y-intersect (Midline) at: (0, 728)
  • Max (right after Y-axis) at: (365/4, 780)
• Since 750 is not yet the Max 780, so it's EARLIER than 365/4 aka.91.25.
• Filter out the 157 which is later than 91.25, so the 24.57 is the answer.

EXAMPLE

Solve:

• Set equation to I(t)=5.2
• Simplify equation to:
• Get θ ≃ 0.9273 Rad
• Get First solution:
• Use the trig identity for second solution: cos(θ) = cos(2π−θ).
• The answer for SECOND TIME it hits 5.2 is 311 days.

Inverse trig function all possible solutions

Example

Find all possible solutions: cos(x)=0.15

Solve: