# Study Sheet

# Unit 1: Functions

## Words to know:

- `linear relation`
- `quadratic relation`
- `vertex of a parabola`
- `line of best fit`
- `axis of symmetry of a parabola`
- `intercepts`


- ```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   

  
## 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.



## 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.   


## 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">    

### 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```
- It shows the relationship between the x and y values.
- Use `Finite differences` to figure out if its quadraic or linear:
    - If the `first difference` is constant, then its linear. (degree of 1)
    - If the `second difference` is constant, then its quadratic. (degree of 2)

- This is a linear function
 
    |x |y |First difference|
    |:-|:-|:---------------|
    |-3|5|$`\cdots`$|
    |-2|7|5-7 = 2|
    |-1|9|7-9 = 2|
    |0|11|9-11 = 2|
    |1|13|11-13 = 2|
    |2|15|15-13 =2|
    - 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.

- This is a quadractic function

    |x |y |First difference|Second difference|
    |:-|:-|:---------------|:----------------|
    |5|9|$`\cdots`$|$`\cdots`$|
    |7|4|9-4 = 5|$`\cdots`$|
    |9|1|4-1 = 3|5-3 = 2|
    |11|0|1-0 = 1|3 - 1 = 2|
    |13|1|0-1 = -1|1 -(-1) = 2|
    
    - 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.






## 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


### 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: $`y = ax^2 + bx + c`$    
 $`
 D = b^2 - 4ac 
 \begin{cases}
 \text{2 distinct real solutions}, & \text{if } D > 0 \\
 \text{1 real solution}, & \text{if } D = 0 \\
 \text{no real solutions}, &  \text{if } D < 0 
 \end{cases}
 `$

- <img src="https://image.slidesharecdn.com/thediscriminant-160218001000/95/the-discriminant-5-638.jpg?cb=1455754224" width="500">


### Tips
- Read the questions carefully and model the system of equations correctly   
- Be sure to name your equations   
- Label your lines   


## Definitions
 - `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).
 - `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.
 - `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 $`\{x \in \mathbb{R}\} `$
 - `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.
 - `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 $`x`$.
- `Transformations`: transformation are operations performed on functions to change the position or shape of the associated curves or lines.

## Working with Function Notation
- 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 $`f(3) = 2(3)^2+3(3)+5`$.
- We can also represent new functions, the letter inside the brackets is simply a variable, we can change it.
    - Given the example $`g(x) = 2x^2+3x+x`$, if we want $`g(m)`$, we simply do $`g(m) = 2m^2+3m+m`$.

## Vertex Form
- `Vertex from`: $`f(x) = a(x-h)^2 + k`$. 
    - $`(-h, k)`$ is the coordinates of the vertex 

## Axis of symmetry
- $`x = -h`$
- Example:
    - $`f(x) = 2(x-3)^2+7`$
    - $`x = +3`$
    - <img src="https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/axis-of-symmetry-of-a-parabola/ex2.gif" width="300">
     

## Direction of openning $`\pm a`$
- 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.
- $`
    \text{Opening} = 
    \begin{cases}
    \text{if } a > 0, & \text{opens up} \\
    \text{if } a < 0, & \text{opens down}
    \end{cases} 
  `$
- Examples
    - $`f(x) = -5x^2`$ opens down, sad face.
    - $`f(x) = 4(x-5)^2+7`$ opens up, happy face.

## Vertical Translations $`\pm k`$
- $`
  \text{Direction} = 
  \begin{cases}
  \text{if } k > 0, & \text{UP }\uparrow \\
  \text{if } k < 0, & \text{DOWN } \downarrow
  \end{cases}
 `$

## Horizontal Translations $`\pm h`$
- $`
  \text{Direction} = 
  \begin{cases}
  \text{if } -h > 0, & \text{shift to the right} \\
  \text{if } -h < 0, & \text{shift to the left}
  \end{cases}
 `$

- $`f(x) = 1(x-4)^2`$
    -   $`\uparrow`$ congruent to $`f(x) = x^2`$ 
    -   
## Vertical Stretch/Compression   
- $`|a|\leftarrow`$: absolute bracket.
    - simplify and become positive
- $` 
  \text{Stretch/Compression} = 
  \begin{cases}
  \text{if } |a| > 1, & \text{stretch by a factor of } a \\
  \text{if } 0 < |a| < 1, & \text{compress by a factor of } a
  \end{cases}
 `$
    - (Multiply all the y-values from $`y = x^2`$ by a)
    - (Not congruent to $`f(x) = x^2`$)
- Example of stretching
    - $`f(x) = 2x^2`$
        -Vertically stretch by a factor of 2
    - |x |y |
      |:-|:-|
      |-3|9`(2)` = 18|
      |-2|4`(2)` = 8|
      |-1|1`(2)`= 2|
      |0|0`(2)` = 0|
      |1|1`(2)` = 2|
      |2|4`(2)`= 8|
      |3|9`(2)` = 18|

    - All y-values from $`f(x) =x^2`$ are now multiplied by 2 to create $`f(x)=2x^2`$

- Example of compression
   - $`f(x) = \frac{1}{2}x^2`$
        - Verticallyc ompressed by a factor of $`\frac{1}{2}`$
   - |x |y |
     |:-|:-|
     |-3|9$`(\frac{1}{2})`$ = 4.5|
     |-2|4$`(\frac{1}{2})`$ = 2|
     |-1|1$`(\frac{1}{2})`$ = $`\frac{1}{2}`$|
     |0|0$`(\frac{1}{2})`$ = 0|
     |1|1$`(\frac{1}{2})`$ = 1|
     |2|4$`(\frac{1}{2})`$= $`\frac{1}{2}`$|
     |3|9$`(\frac{1}{2})`$ = 4.5|
    
    - All y-values from $`f(x) = x^2`$ are now multiplied by $`\frac{1}{2}`$ to create $`f(x) = \frac{x^2}{2}`$