diff --git a/Grade 9/Math.md b/Grade 9/Math.md index 651c255..e43cd6d 100644 --- a/Grade 9/Math.md +++ b/Grade 9/Math.md @@ -53,7 +53,7 @@ >> | 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 >> a2 + b2 = c2 >> @@ -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: ->>> = 3⁄4 ÷ 2⁄12 [Reduce to lowest terms] ->>> = 3⁄4 ÷ 1⁄7 [Multiply by reciprocal] ->>> = 3⁄4 × 7 ->>> = 21⁄4 [Leave as an improper fraction] +>>> = [Reduce to lowest terms] +>>> = [Multiply by reciprocal] +>>> = × 7 +>>> = [Leave as an improper fraction] >> ### Shortcut for multiplying fractions >>> cross divide to keep your numbers small @@ -93,7 +93,7 @@ >>> |Rational Exponents|an/m = (m√a)n|165/4 = (4√16)5 = 25| >>> **Note:** ->>> Standard --> Expanded Form +>>> Exponential Form --> Expanded Form >>> 64 = 6 × 6 × 6 × 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 - ->> - ->> ```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```: ∪ ->> if 2 sets have union, all the elements belong to any of the set. This is known as ```or``` ->> ```Intersection```: ∩ ->> 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∈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: >> √x2 = ± 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 = 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** - + > ## Tips >> ```Absolute Values``` can have 2 solutions >> ```Quadratics``` can also have 2 solutions @@ -292,10 +258,22 @@ > > 7. ``` Isosceles Triangle Theorem``` (ITT) -> - The base angles in any isosceles triangle are equal +> - 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 + + + + +> 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 @@ -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 = y2-y1/x2-x1** +>> ```Slope```: The measure of the steepness of a line - ```rise / run``` or ```rate of change y / rate of change x``` +>> ```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)** @@ -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 × slope2 = -1 >> Notation: m⊥ >> @@ -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 ->> +>> + +>> **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``` @@ -486,6 +475,32 @@ >> ### 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