From 1c74d2b0422f7beea04897eb3b41a9638bf6bf31 Mon Sep 17 00:00:00 2001 From: James Su Date: Thu, 18 Apr 2019 23:59:38 +0000 Subject: [PATCH] Update Final_Exam_Study_Sheet.md --- Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md | 124 ++++++++++-------- 1 file changed, 67 insertions(+), 57 deletions(-) diff --git a/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md b/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md index 14a0c1b..0b78b55 100644 --- a/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md +++ b/Grade 9/Math/MPM1DZ/Final_Exam_Study_Sheet.md @@ -136,9 +136,9 @@ ## 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. +- 555 (**Exact**) or $`5.55 \times 10^2`$ (**3 significant figures**). +- In scientific notation, values are written in the form $`a(10^n)`$, where $`a`$ is a number within 1 and 10 and $`n`$ is any integer. +- Some examples include the following: $`5.4 \times 10^3, 3.0 \times 10^2`$, and $`4.56 \times 10^{-4}`$. - When the number is smaller than 1, a negative exponent is used, when the number is bigger than 10, a positve exponent is used - @@ -340,20 +340,20 @@ x, & \text{if } x > 0\\ ## 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|| + |Rectangle|```Area```: $`lw`$
```Perimeter```: $`2(l+w)`$|| + |Triangle|```Area```: $`\frac{bh}{2}`$
```Perimeter```: $`a+b+c`$|| + |Circle|```Area```: $`πr^2`$
```Circumference```: $`2πr`$ or $`πd`$|| + |Trapezoid|```Area```: $` \frac{(a+b)h}{2}`$
```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|| + |Rectangular Prism|```Volume```: $`lwh`$
```SA```: $`2(lw+lh+wh)`$|| + |Square Based Pyramid|```Volume```: $`\frac{1}{3} b^2 h`$
```SA```: $`2bs+b^2`$|| + |Sphere|```Volume```: $`\frac{4}{3} πr^3`$
```SA```: $`4πr^2`$|| + |Cone|```Volume```: $` \frac{1}{3} πr^2 h`$
```SA```: $`πrs+πr^2`$|| + |Cylinder|```Volume```: $`πr^2h`$
```SA```: $`2πr^2+2πh`$|| + |Triangular Prism|```Volume```: $`ah+bh+ch+bl`$
```SA```: $` \frac{1}{2} blh`$|| ## Optimization (For Maximimizing Area/Volume, or Minimizing Perimeter/Surface Area) @@ -362,18 +362,18 @@ x, & \text{if } x > 0\\ |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| + |4-sided rectangle|A rectangle must be a square to maximaze the area for a given perimeter. The length is equal to the width
$`A = lw`$
$`A_{max} = (w)(w)`$
$`A_{max} = w^2`$|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|$`l = 2w`$
$`A = lw`$
$`A_{max} = 2w(w)`$
$`A_{max} = 2w^2`$|$`l = 2w`$
$`P = l+2w`$
$`P_{min} = 2w+2w`$
$`P_{min} = 4w`$| ### 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
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| + |Cylinder(closed-top)|The cylinder must be similar to a cube where $`h = 2r`$
$`V = πr^2h`$
$`V_{max} = πr^2(2r)`$
$`V_{max} = 2πr^3`$|The cylinder must be similar to a cube where $`h = 2r`$
$`SA = 2πr^2+2πrh`$
$`SA_{min} = 2πr^2+2πr(2r)`$
$`SA_{min} = 2πr^2+4πr^2`$
$`SA_{min} = 6πr^2`$| + |Rectangular Prism(closed-top)|The prism must be a cube,
where $`l = w = h`$
$`V = lwh`$
$`V_{max} = (w)(w)(w)`$
$`V_{max} = w^3`$|The prism must be a cube,
where $`l = w = h`$
$`SA = 2lh+2lw+2wh`$
$`SA_{min} = 2w^2+2w^2+2w^2`$
$`SA_{min} = 6w^2`$| + |Cylinder(open-top)|$`h = r`$
$`V = πr^2h`$
$`V_{max} = πr^2(r)`$
$`V_{max} = πr^3`$|$`h = r`$
$`SA = πr^2+2πrh`$
$`SA_{min} = πr^2+2πr(r)`$
$`SA_{min} = πr^2+2πr^2`$
$`SA_{min} = 3πr^2`$| + |Square-Based Rectangular Prism(open-top)|$`h = \frac{w}{2}`$
$`V = lwh`$
$`V_{max} = (w)(w)(\frac{w}{2})`$
$`V_{max} = \frac{w^3}{2}`$|$`h = \frac{w}{2}`$
$`SA = w^2+4wh`$
$`SA_{min} = w^2+4w(\frac{w}{2})`$
$`SA_{min} = w^2+2w^2`$
$`SA_{min} = 3w^2`$| ## Labelling - Given any polygons, labelling the vertices must always: @@ -420,10 +420,10 @@ x, & \text{if } x > 0\\ ## 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)** +- ```Slope Formula```: $`m = \frac{y_2 - y_1}{x_2 - x_1}`$ +- ```Standard Form```: $`ax + by + c = 0, a \isin \mathbb{Z}, b \isin \mathbb{Z}, c \isin \mathbb{Z}`$ (must be integers and $`a`$ must be positive) +- ```Y-intercept Form```: $`y = mx + b`$ +- ```Point-slope Form```: $`y_2-y_1 = m(x_2-x_1)`$ - The slope of a vertical lines is undefined - The slope of a horizontal line is 0 - Parallel lines have the ```same slope``` @@ -432,25 +432,25 @@ x, & \text{if } x > 0\\ ## Relations - A relation can be described using 1. Table of Values (see below) - 2. Equations (y = 3x + 5) + 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. +- 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⊥ +- Notation: $`m_\perp`$ - ## 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). +- ```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. @@ -525,18 +525,22 @@ x, & \text{if } x > 0\\ - ## 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)``` +- 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: $`y = ax^2 + bx + c`$ + $` + D = b^2 - 4ac + \begin{cases} + \text{2 distinct real solutions}, & \text{if } D > 0 \\ + \text{1 real solution}, & \text{if } D = 0 \\ + \text{no real solutions}, & \text{if } D < 0 + \end{cases} + `$ - ## Solving Linear-Quadratic Systems - To find the point of intersection, do the following: - 1. Isolate both equations for ```y``` + 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 @@ -547,7 +551,7 @@ x, & \text{if } x > 0\\ - - There are 3 possible cases -- In addition, to determine the number of solutions, you the Discriminant formula **D = b2 - 4ac** +- In addition, to determine the number of solutions, you the Discriminant formula $`D = b^2 - 4ac`$ # Ways to solve Systems of Equations @@ -559,14 +563,18 @@ x, & \text{if } x > 0\\ 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``` + - 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 @@ -576,7 +584,7 @@ x, & \text{if } x > 0\\ 2x + 3y = 10 (1) 4x + 3y = 14 (2) ``` - We can then use elimination + - We can then use elimination ``` 4x + 3y = 14 2x + 3y = 10 @@ -584,10 +592,12 @@ x, & \text{if } x > 0\\ 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``` + - 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 @@ -598,20 +608,20 @@ x, & \text{if } x > 0\\ - | |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| + |Shade the region ```above``` the line|$`y > mx + b`$|$`y \ge mx + b`$| + |Shade the region ```below``` the line|$`y < mx + b`$|$`y \le mx + b`$| - ## If - - |x > a
x ≥ a| + - |$`x > a`$
$`x \ge a`$| |:------------------| - - |shade the region on the **right**| + |shade the region on the **right**| - ## If - - |x < a
x ≤ a| - - |:------------------| - - |shade the region on the **left**| + - |$`x < a`$
$`x \le a`$| + |:------------------| + |shade the region on the **left**| - Step 1. change all inequalities to ```y-intercept form``` - Step 2. graph the line