From 8a635a2b6fea82e2bc8ce37b03e6d772ac12516a Mon Sep 17 00:00:00 2001 From: James Su Date: Wed, 4 Sep 2019 23:37:33 +0000 Subject: [PATCH] Add new file --- ...nalytical Geometry and Linear Relations.md | 96 +++++++++++++++++++ 1 file changed, 96 insertions(+) create mode 100644 Grade 9/Math/MPM1DZ/Unit 5: Analytical Geometry and Linear Relations.md diff --git a/Grade 9/Math/MPM1DZ/Unit 5: Analytical Geometry and Linear Relations.md b/Grade 9/Math/MPM1DZ/Unit 5: Analytical Geometry and Linear Relations.md new file mode 100644 index 0000000..0011cff --- /dev/null +++ b/Grade 9/Math/MPM1DZ/Unit 5: Analytical Geometry and Linear Relations.md @@ -0,0 +1,96 @@ +# 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```: $`m = \frac{y_2 - y_1}{x_2 - x_1}`$ +- ```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``` +- 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: $`m_\perp`$ +- + + +## 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 $`y`$. +- ```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:** + + 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``` + + | 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 \ No newline at end of file