## Function Transformations

How move and resize the graphs of functions?

Example

*f(x) = x*^{2}^{}

Table of Contents

## How move the graphs of functions?

- Add a constant to the y-value will move it up or down
- Add a constant to the x-value will move it left or right

Examples

Function | c > 0 | c < 0 |

f(x) = x^{2} g(x) = x^{2} + c | ||

f(x) = x^{2} g(x) = (x + c)^{2} |

To put it simply:

*add to y to go high**add to x to go left*

## How resize the graphs of functions?

- multiplying the whole function by a constant will stretch or compress it in the y-direction
- multiplying x by a constant will stretch or compress it in the x-direction

Examples

Function | c > 1 | 0 < c < 1 |

f(x) = x^{2} g(x) = cx^{2} | ||

f(x) = x^{2} g(x) = (cx)^{2} |

Note:

*for the y-direction, bigger values cause more stretch.**for the x-direction, bigger c cause more compression.*

## How flip the graphs of functions?

- Multiplying the whole function by −1 will flip it upside down
- Multiplying the x-value by −1 will flip it upside down

Function | Graph |

f(x) = x^{2}g(x) = -x^{2} | |

f(x) = x^{2}g(x) = (-x)^{2} |

## Summary

y = f(x) + c | c > 0 moves it upc < 0 moves it down |

y = f(x + c) | c > 0 moves it leftc < 0 moves it right |

y = cf(x) | c > 1 stretches it in the y-direction0 < c < 1 compresses it |

y = f(cx) | c > 1 stretches it in the x-direction0 < c < 1 stretches it |

y = -f(x) | Reflects it about x-axis |

y = f(-x) | Reflects it about y-axis |

## All transformation in one go

To put it simply, use

*a f(b(x + c)) + d*

Symbol | Detail |

a | |a| > 1 stretches |a| < 1 compresses a < 0 flips the graph upside down |

b | |b| > 1 compresses |b| < 1 stretches b < 0 flips the graph left-right |

c | c < 0 shifts to the right c > 0 shifts to the left |

d | d > 0 shifts upward d < 0 shifts downward |