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# Study Sheet
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# Unit 1
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# Unit 1: Functions
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## Words to know:
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- `axis of symmetry of a parabola`
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- `intercepts`
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- ```Linear Relation```: A relation which a single straight line can be drawn through every data point and the first differences are constant
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- ```Non - Linear Relation```: A single smooth curve can be drawn through every data point and the first differences are not constant
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## Relations
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- A relation can be described using
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1. Table of Values (see below)
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2. Equations $`(y = 3x + 5)`$
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3. Graphs (Graphing the equation)
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4. Words
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- When digging into the earth, the temperature rises according to the
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- following linear equation: $`t = 15 + 0.01 h`$. $`t`$ is the increase in temperature in
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- degrees and $`h`$ is the depth in meters.
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## Definitions
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- ```Parallel```: 2 lines with the same slope
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- ```Perpendicular```: 2 lines with slopes that are the negative reciprocal to the other. They form a 90 degree angle where they meet.
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- ```Domain```: The **ordered** set of all possible values of the independent variable $`x`$.
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- ```Range```: The **ordered** set of all possible values of the dependent variable $`y`$.
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- ```Continous Data```: A data set that can be broken into smaller parts. This is represented by a ```Solid line```.
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- ```Discrete Data```: A data set that **cannot** be broken into smaller parts. This is represented by a ```Dashed line```.
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- ```First Difference```: the difference between 2 consecutive y values in a table of values which the difference between the x-values are constant.
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- ```Collinear Points```: points that line on the same straight line
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## Variables
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- ```Independent Variable```: A Variable in a relation which the values can be chosen or isn't affected by anything.
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- ```Dependent Varaible```: A Variable in a relation which is **dependent** on the independent variable.
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## Scatterplot and Line of Best Fit
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- A scatterplot graph is there to show the relation between two variables in a table of values.
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- A line of best fit is a straight line that describes the relation between two variables.
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- 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.
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- <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/line%20of%20best%20fit-eyeball/lineofbestfit-e-1.gif" width="300">
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### How To Determine the Equation Of a Line of Best Fit
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1. Find two points **```ON```** the ```line of best fit```
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2. Determine the ```slope``` using the two points
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3. Use ```point-slope form``` to find the equation of the ```line of best fit```
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## Table of values
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- To find first differences or any points on the line, you can use a ```table of values```
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- It shows the relationship between the x and y values.
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- Use `Finite differences` to figure out if its quadraic or linear:
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- If the `first difference` is constant, then its linear. (degree of 1)
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- If the `second difference` is constant, then its quadratic. (degree of 2)
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- This is a linear function
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|x |y |First difference|
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|:-|:-|:---------------|
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|-3|5|$`\cdots`$|
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|-2|7|5-7 = 2|
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|-1|9|7-9 = 2|
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|0|11|9-11 = 2|
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|1|13|11-13 = 2|
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|2|15|15-13 =2|
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- The difference between the first and second y values are the same as the difference between the third and fourth. The `first difference` is constant.
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- This is a quadractic function
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|x |y |First difference|Second difference|
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|:-|:-|:---------------|:----------------|
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|5|9|$`\cdots`$|$`\cdots`$|
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|7|4|9-4 = 5|$`\cdots`$|
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|9|1|4-1 = 3|5-3 = 2|
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|11|0|1-0 = 1|3 - 1 = 2|
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|13|1|0-1 = -1|1 -(-1) = 2|
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- The difference between the differences of the first and second y values are the same as the difference of the difference between the thrid and fourth. The `second difference` is constant.
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## Tips
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- Label your graph correctly, the scales/scaling and always the ```independent variable``` on the ```x-axis``` and the ```dependent variable``` on ```y-axis```
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- Draw your ```Line of Best Fit``` correctly
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- Read the word problems carefully, and make sure you understand it when graphing things
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- Sometimes its better not to draw the shape, as it might cloud your judgement (personal exprience)
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- Label your lines
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### Number of Solutions
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- <img src="https://lh5.googleusercontent.com/wqYggWjMVXvWdY9DiCFYGI7XSL4fXdiHsoZFkiXcDcE93JgZHzPkWSoZ6f4thJ-aLgKd0cvKJutG6_gmmStSpkVPJPOyvMF4-JcfS_hVRTdfuypJ0sD50tNf0n1rukcLBNqOv42A" width="500">
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## Discriminant
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- The discriminant determines the number of solutions (roots) there are in a quadratic equation. $`a, b , c`$ are the
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- coefficients and constant of a quadratic equation: $`y = ax^2 + bx + c`$
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$`
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D = b^2 - 4ac
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\begin{cases}
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\text{2 distinct real solutions}, & \text{if } D > 0 \\
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\text{1 real solution}, & \text{if } D = 0 \\
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\text{no real solutions}, & \text{if } D < 0
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\end{cases}
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`$
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- <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">
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### Tips
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- Read the questions carefully and model the system of equations correctly
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- Be sure to name your equations
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- Label your lines
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## Definitions
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- `Function`: a relation which there is only one value of the dependent variable for each value of the independent variable (i.e, for every x-value, there is only one y-value).
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- `Vertical-line test`: a test to determine whether the graph of a relation is a function. The relation is not a function if at least one vertical line drawn through the graph of the relation passes through two or more points.
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- `Real numbers`: the set of real numbers is the set of all decimals - positive, negative and 0, terminating and non-terminating. This statement is expressed mathematically with the set notation $`\{x \in \mathbb{R}\} `$
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- `Degree`: the degree of a polynomial with a single varible, say $`x`$, is the value of the highest exponent of the variable. For example, for the polynomial $`5x^3-4x^2+7x-8`$, the highest power or exponent is 3; the degree of the polynomial is 3.
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- `Function notation`: $`(x, y) = (x f(x))`$. $`f(x)`$ is called function notation and represents the value of the dependent variable for a given value of the independent variable $`x`$.
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- `Transformations`: transformation are operations performed on functions to change the position or shape of the associated curves or lines.
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## Working with Function Notation
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- Given an example of $`f(x) = 2x^2+3x+5`$, to get $`f(3)`$, we substitute the 3 as $`x`$ into the function, so it now becomses $`f(3) = 2(3)^2+3(3)+5`$.
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- We can also represent new functions, the letter inside the brackets is simply a variable, we can change it.
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- Given the example $`g(x) = 2x^2+3x+x`$, if we want $`g(m)`$, we simply do $`g(m) = 2m^2+3m+m`$.
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## Vertex Form
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- `Vertex from`: $`f(x) = a(x-h)^2 + k`$.
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- $`(-h, k)`$ is the coordinates of the vertex
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## Axis of symmetry
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- $`x = -h`$
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- Example:
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- $`f(x) = 2(x-3)^2+7`$
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- $`x = +3`$
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- <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/axis-of-symmetry-of-a-parabola/ex2.gif" width="300">
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## Direction of openning $`\pm a`$
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- Given a quadratic in the from $`f(x) = ax^2+bx+c`$, if $`a > 0`$, the curve is a happy face, a smile. If $`a < 0`$, the curve is a sad face, a sad frown.
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- $`
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\text{Opening} =
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\begin{cases}
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\text{if } a > 0, & \text{opens up} \\
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\text{if } a < 0, & \text{opens down}
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\end{cases}
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`$
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- Examples
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- $`f(x) = -5x^2`$ opens down, sad face.
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- $`f(x) = 4(x-5)^2+7`$ opens up, happy face.
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## Vertical Translations $`\pm k`$
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- $`
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\text{Direction} =
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\begin{cases}
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\text{if } k > 0, & \text{UP }\uparrow \\
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\text{if } k < 0, & \text{DOWN } \downarrow
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\end{cases}
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`$
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## Horizontal Translations $`\pm h`$
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- $`
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\text{Direction} =
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\begin{cases}
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\text{if } -h > 0, & \text{shift to the right} \\
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\text{if } -h < 0, & \text{shift to the left}
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\end{cases}
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`$
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- $`f(x) = 1(x-4)^2`$
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- $`\uparrow`$ congruent to $`f(x) = x^2`$
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-
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## Vertical Stretch/Compression
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- $`|a|\leftarrow`$: absolute bracket.
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- simplify and become positive
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- $`
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\text{Stretch/Compression} =
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\begin{cases}
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\text{if } |a| > 1, & \text{stretch by a factor of } a \\
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\text{if } 0 < |a| < 1, & \text{compress by a factor of } a
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\end{cases}
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`$
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- (Multiply all the y-values from $`y = x^2`$ by a)
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- (Not congruent to $`f(x) = x^2`$)
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- Example of stretching
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- $`f(x) = 2x^2`$
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-Vertically stretch by a factor of 2
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- |x |y |
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|:-|:-|
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|-3|9`(2)` = 18|
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|-2|4`(2)` = 8|
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|-1|1`(2)`= 2|
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|0|0`(2)` = 0|
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|1|1`(2)` = 2|
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|2|4`(2)`= 8|
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|3|9`(2)` = 18|
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- All y-values from $`f(x) =x^2`$ are now multiplied by 2 to create $`f(x)=2x^2`$
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- Example of compression
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- $`f(x) = \frac{1}{2}x^2`$
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- Verticallyc ompressed by a factor of $`\frac{1}{2}`$
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- |x |y |
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|:-|:-|
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|-3|9$`(\frac{1}{2})`$ = 4.5|
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|-2|4$`(\frac{1}{2})`$ = 2|
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|-1|1$`(\frac{1}{2})`$ = $`\frac{1}{2}`$|
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|0|0$`(\frac{1}{2})`$ = 0|
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|1|1$`(\frac{1}{2})`$ = 1|
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|2|4$`(\frac{1}{2})`$= $`\frac{1}{2}`$|
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|3|9$`(\frac{1}{2})`$ = 4.5|
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- All y-values from $`f(x) = x^2`$ are now multiplied by $`\frac{1}{2}`$ to create $`f(x) = \frac{x^2}{2}`$
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