03: Various Types of Functions 

# Parent Functions

\
## Linear Function

f(x) = x

| x | y |
|----|----|
| -2 | -2 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |

Domain: {x E R}

Range: {y E R}

\
## Quadratic Function

f(x) = x²

| x | y |
|----|----|
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |

Domain: {x E R}

Range: {y E R/0 ≤ y}

\
## Square Root Function

f(x) = √x

| x | y |
|----|----|
| 0 | 0 |
| 1 | 1 |
| 4 | 2 |

Domain: {x E R/0 ≤ x}

Range: {y E R/0 ≤ y}

\
## Reciprocal Function

f(x) = 1/x

| x | y |
|----|----|
| -2 | -1/2 |
| -1 | -1 |
| -0.5 | -2 |
| 0.5 | 2 |
| 1 | 1 |
| 2 | 1/2 |

Domain: {x E R/x ≠ 0}

Range: {y E R/y ≠ 0}

Asymptote: x = 0, y = 0

\
## Absolute Value Function

f(x) = |x|

| x | y |
|----|----|
| -2 | 2 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |

Domain: {x E R}

Range: {y E R/0 ≤ y}

\
## Cubic Function

f(x) = x³

| x | y |
|----|----|
| -2 | -8 |
| -1 | -1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 8 |

Domain: {x E R}

Range: {y E R}

\
# Transformations of Parent Functions

> **Transformed functions: @@f(x) = a(k(x-d)) + c@@**

* Vertical Stretch: ==a==
* By a factor of….
* If negative, reflection in the x axis
* Horizontal Stretch: ^^k^^
* Always **1/k (flipped)**
* By a factor of….
* If negative, reflection in the y axis
* Vertical Translation: ==c==
* if positive, moves up
* If negative, moves down
* Horizontal Translation: ^^d^^
* Always the **opposite sign** of what it is in the brackets (sign is flipped)
* If positive in bracket (so negative alone), then it moves left (