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>> | a ∩ b | Intersection (and) |
> ## Pythgorean Theorem
>> let a be the adjecant and b be the opposite, and c be the hypotenuse.
>> 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
>> a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup>
>> <img src="https://docs.google.com/drawings/u/1/d/sGjyHDIs-wHWzppHAGdIpEA/image?w=162&h=70&rev=1&ac=1&parent=1ZIXKcDk3LBlgPK2EoUV04c0G1LZotrtfgVhJTooO1zA" width="200">
@ -68,10 +68,10 @@
>> To Multiply rationals, first reduce the fraction to their lowest terms, then multiply numerators and denominators
>> To Divide rationals, multiply them by the reciprocal
>> ### Example Simplify Fully:
>>> = <sup>3</sup>&frasl;<sub>4</sub> &divide; <sup>2</sup>&frasl;<sub>12</sub> [Reduce to lowest terms]
>>> = <sup>3</sup>&frasl;<sub>4</sub> &divide; <sup>1</sup>&frasl;<sub>7</sub> [Multiply by reciprocal]
>>> = <sup>3</sup>&frasl;<sub>4</sub> &times; 7
>>> = <sup>21</sup>&frasl;<sub>4</sub> [Leave as an improper fraction]
>>> = <a href="https://www.codecogs.com/eqnedit.php?latex==&space;\frac{3}{4}&space;\div&space;\frac{2}{14}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?=&space;\frac{3}{4}&space;\div&space;\frac{2}{14}" title="= \frac{3}{4} \div \frac{2}{14}" /></a>[Reduce to lowest terms]
>>> = <a href="https://www.codecogs.com/eqnedit.php?latex=\frac{3}{4}&space;\div&space;\frac{1}{7}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{3}{4}&space;\div&space;\frac{1}{7}" title="\frac{3}{4} \div \frac{1}{7}" /></a> [Multiply by reciprocal]
>>> = <a href="https://www.codecogs.com/eqnedit.php?latex=\frac{3}{4}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{3}{4}" title="\frac{3}{4}" /></a> &times; 7
>>> = <a href="https://www.codecogs.com/eqnedit.php?latex=\frac{21}{4}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\frac{21}{4}" title="\frac{21}{4}" /></a> [Leave as an improper fraction]
>> ### Shortcut for multiplying fractions
>>> cross divide to keep your numbers small
@ -93,7 +93,7 @@
>>> |Rational Exponents|a<sup>n/m</sup> = (<sup>m</sup>&radic;a)<sup>n</sup>|16<sup>5/4</sup> = (<sup>4</sup>&radic;16)<sup>5</sup> = 2<sup>5</sup>|
>>> **Note:**
>>> Standard --> Expanded Form
>>> Exponential Form --> Expanded Form
>>> 6<sup>4</sup> = 6 &times; 6 &times; 6 &times; 6
>> ## Scientific Notation
@ -188,26 +188,16 @@
>>> 1. Expand and simplify both sides
>>> 2. Isolate using reverse order of operations
>>> 3. Check the solution by plugging the variable back into the equation and check if the ```left side``` equals the ```right side```
> ## Venn Diagrams
>> <img src="https://d2slcw3kip6qmk.cloudfront.net/marketing/blog/2017Q3/Venn-diagram-symbols-and-notation/VDIntersections.png" width="400">
>> ```Set```: a collection of elements, O (the circle)
>> ```Universal Set```: This is a collection of all the elements that you are interested in. Use ```{}``` bracket to write the set inside the rectangle
>> ```Union```: &cup;
>> if 2 sets have union, all the elements belong to any of the set. This is known as ```or```
>> ```Intersection```: &cap;
>> if 2 sets have an intersection, they are elements belonging to both sets. This is known as ```and```
>> ```Set Notation```:
>> A notation that represents the collection of numbers. It is written in this form x = {x|x&isin;R}
>> ```Absolute Value```
>>> There are 2 cases. For this sort of equation, you must split the equation into 2 separate equations. One of the
>>> equations will have the absolute bracket be positive while the other negative.
>>> Absolute values are written in the form ```| x |```
>>> where
>>> if x > 0, | x | = x
>>> if x == 0, | x | = 0
>>> if x < 0, | x | = -x
>
> ## 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
@ -225,31 +215,7 @@
>> Note:
>> &radic;x<sup>2</sup> = &plusmn; x (There are 2 possible solutions)
>> ```Distrubutive Property``` - This is opening the bracket. a(x+y) = ax+ay
>> ## Discriminant
>>> The discriminant determines the number of solutions (roots) there are in a quadratic equation. ```a```, ```b```, ```c``` are the
>>> coefficients and constant of a quadratic equation: ```y = ax<sup>2</sup> + bx + c```
>>> D = b<sup>2</sup> - 4ac
>>> D > 0 ```(2 distinct real solutions)```
>>> D = 0 ```(1 real solution)```
>>> D < 0 ```(no real solutions)```
>>> <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">
> ## Solving Linear-Quadratic Systems
>> To find the point of intersection, do the following:
>> 1. Isolate both equations for ```y```
>> 2. Set the equations equal to each other by ```subsitution``` Equation 1 = Equation 2
>> 3. Simplify and put everything on one side and equal to zero on the other side
>> 4. Factor
>> 5. Use zero-product property to solve for all possible x-values
>> 6. Subsitute the x-values to one of the original equations to solve for all y-values
>> 7. State a conclusion / the solution
> <img src = "https://lh5.googleusercontent.com/AJxSjT24kwneM_UH6kehfX-7AnzVewTJIk6v02aXOZ84veou2xNyBMPmhGSXWNhvhJfZT-wwHSlDNvbsfeHzjpGSuXMOohoIvaS2u0saoO1BZTRV3xNVobdoWytLhkVl0CkEaIiQ" width ="500">
> There are 3 possible cases
> In addition, to determine the number of solutions, you the Discriminant formula **D = b<sup>2</sup> - 4ac**
> ## Tips
>> ```Absolute Values``` can have 2 solutions
>> ```Quadratics``` can also have 2 solutions
@ -292,10 +258,22 @@
> <img src="https://www.katesmathlessons.com/uploads/1/6/1/0/1610286/exterior-angle-theorem-diagram-picture_orig.png" width="300">
> 7. ``` Isosceles Triangle Theorem``` (ITT)
> - The base angles in any isosceles triangle are equal
> - The base angles in any isosceles triangle are equal
> <img src="http://www.assignmentpoint.com/wp-content/uploads/2016/06/isosceles-triangle-theorem.jpg" width="400">
> 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
<img src="https://i.ytimg.com/vi/tmRpwCM1K1o/maxresdefault.jpg" width="500">
> 9. ```Exterior Angles of a Convex Polygon```
> - The sum of the exterior angle of any convex polygon is always ```360 degrees```
<img src="https://image.slidesharecdn.com/findanglemeasuresinapolygon-110307143453-phpapp02/95/find-angle-measures-in-a-polygon-11-728.jpg?cb=1299508555" width="400">
> ## Properties of Quadrilaterals
>> Determine the shape using the properties of it
@ -392,8 +370,8 @@
> ```Linear Relation```: A relation which a single straight line can be drawn through every data point and the first differences are constant
> ```Non - Linear Relation```: A single smooth curve can be drawn through every data point and the first differences are not constant
> ## Slope and Equation of Line
>> ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```change in y / change in x```
>> ```Slope Formula```: **m = y<sub>2</sub>-y<sub>1</sub>/x<sub>2</sub>-x<sub>1</sub>**
>> ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```rate of change y / rate of change x```
>> ```Slope Formula```: **m = (y<sub>2</sub>-y<sub>1</sub>)/(x<sub>2</sub>-x<sub>1</sub>)**
>> ```Standard Form```: **ax + by + c = 0**, a&isin;Z, b&isin;Z, c&isin;Z (must be integers and ```a``` must be positive)
>> ```Y-intercept Form```: **y = mx + b**
>> ```Point-slope Form```: **y<sub>2</sub>-y<sub>1</sub> = m(x<sub>2</sub>-x<sub>1</sub>)**
@ -412,8 +390,8 @@
>> following linear equation: t = 15 + 0.01 h. **t** is the increase in temperature in
>> degrees and **h** is the depth in meters.
> ## Perpendicular Bisector
>> To find the perpendicular bisector, you will need to fidn the slope and midpoint
> ## Perpendicular Lines
>> To find the perpendicular slope, you will need to find the slope point
>> Formula: slope1 &times; slope2 = -1
>> Notation: m<sub>&perp;</sub>
>> <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/parallel-perpendicular-lines/parallel_perpendicular_lines_1.gif" width="300">
@ -446,13 +424,24 @@
>> You can't go backwards on the x-axis, as you can't go back in time
>> Plot the points accordingly
>> Draw the lines accordingly
>> <img src="https://dryuc24b85zbr.cloudfront.net/tes/resources/6061038/image?width=500&height=500&version=1519313844425" width="400">
>> <img src="https://dryuc24b85zbr.cloudfront.net/tes/resources/6061038/image?width=500&height=500&version=1519313844425" width="400">
>> **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.
>> <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/line%20of%20best%20fit-eyeball/lineofbestfit-e-1.gif" width="300">
>> <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/line%20of%20best%20fit-eyeball/lineofbestfit-e-1.gif" width="300">
>> ### 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```
@ -486,6 +475,32 @@
>> ### Number of Solutions
>>> <img src="https://lh5.googleusercontent.com/wqYggWjMVXvWdY9DiCFYGI7XSL4fXdiHsoZFkiXcDcE93JgZHzPkWSoZ6f4thJ-aLgKd0cvKJutG6_gmmStSpkVPJPOyvMF4-JcfS_hVRTdfuypJ0sD50tNf0n1rukcLBNqOv42A" width="500">
> ## Discriminant
>> The discriminant determines the number of solutions (roots) there are in a quadratic equation. ```a```, ```b```, ```c``` are the
>> coefficients and constant of a quadratic equation: ```y = ax<sup>2</sup> + bx + c```
>> D = b<sup>2</sup> - 4ac
>> D > 0 ```(2 distinct real solutions)```
>> D = 0 ```(1 real solution)```
>> D < 0 ```(no real solutions)```
>> <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">
> ## Solving Linear-Quadratic Systems
>> To find the point of intersection, do the following:
>> 1. Isolate both equations for ```y```
>> 2. Set the equations equal to each other by ```subsitution``` Equation 1 = Equation 2
>> 3. Simplify and put everything on one side and equal to zero on the other side
>> 4. Factor
>> 5. Use zero-product property to solve for all possible x-values
>> 6. Subsitute the x-values to one of the original equations to solve for all y-values
>> 7. State a conclusion / the solution
>> <img src = "https://lh5.googleusercontent.com/AJxSjT24kwneM_UH6kehfX-7AnzVewTJIk6v02aXOZ84veou2xNyBMPmhGSXWNhvhJfZT-wwHSlDNvbsfeHzjpGSuXMOohoIvaS2u0saoO1BZTRV3xNVobdoWytLhkVl0CkEaIiQ" width ="500">
>> There are 3 possible cases
>> In addition, to determine the number of solutions, you the Discriminant formula **D = b<sup>2</sup> - 4ac**
> # Ways to solve Systems of Equations
> 1. Subsitution
> Here we eliminate a variable by subbing in another variable from another equation