# Formulas, graphs & relations » Mutations on formulas and graphs (translating and stretching)

## Contents

1. Translation (addition)2. Stretching (multiplication)

## 1. Translation (addition)

You can change a graph by moving it vertically or horizontally. This is called translation.

### Vertical translation (up/down)

To translate a formula vertically with `q`, you just add + `q` behind the formula. When translating down `q` is negative. To plot the new graph from the old graph is easy. Just read off every value, add `q` and plot the new graph.

*Example*

Below you can see how function `f` (`x`) = `x`^{3} is translated upwards by 4.

The function for the new graph is `g`(`x`) = `x`^{3} + 4.

### Horizontal translation (left/right)

When translating a formula or graph horizontally to the right with `p`, you change every `x` in the old formula with (`x` – `p`).

When translating left `p` is negative. In that case you will get (`x` – (–…)) = (`x` + …).

For the graphs it works the same as above, only now you have to read off every point and you will have to plot the new points `p` to the right.

*Example*

The function `f` (`x`) = 2`x`^{2} + 3`x` is translated 5 units to the left.

The new function is `g`(`x`) = 2(`x` + 5)^{2} + 3(`x` + 5).

By removing brackets you will get:

g(x) = | 2(x + 5)^{2} + 3(x + 5) |

2(x + 5)(x + 5) + 3x + 15 | |

2(x^{2} + 10x + 25) + 3x + 15 | |

2x^{2} + 20x + 50 + 3x + 15 | |

2x^{2} + 23x + 65. |

### Vertical and horizontal

For a translation to the right with `p` and upwards with `q`, you will get translation (`p`, `q`).

For the standard formula `y` = `ax ^{n}` then new formula will be:

`y`=

`a`(

`x`–

`p`)

`+`

^{n}`q`.

## 2. Stretching (multiplication)

You can change a graph by multiplying it with a certain number. This is called stretching.

### Stretching parallel to the `y`-axis

When stretching parallel to the `y`-axis with a factor `a` all outcomes of a formula are multiplied with `a`. The graph will be `a` times further from the `x`-axis.

The new formula can be easily found:

`new formula` = `a`(`old formula`).

Mind the brackets around the old formula!

*Example*

The graph of function `f` (`x`) = `x`^{2} – 1 is stretched parallel to the `y`-axis with factor 2.

All points of the new graph will be twice as far from the `x`-axis.

The function of the new graph will be `g`(`x`) = 2(`x`^{2} – 1) = 2`x`^{2} – 2.

### Stretching parallel to the `x`-axis

When stretching parallel to the `x`-axis with factor `a` you substitute every `x` elke `x` in the function by (1`a``x`).

When plotting the new graph it works the same, now the new points will have to be `a` times further from the `y`-axis.

*Example*

The function `f` (`x`) = 3`x`^{2} + 5`x` – 2 is stretched parallel to the `x`-axis with factor 4.

The new function will be g(x)^{ }= | 3(14x)^{2} + 5(14x) – 2 |

^{ }= | 3(14)^{2}x^{2} + 54x – 2. |

^{ }= | 3 × 116 × x^{2} + 114x – 2. |

^{ }= | 316x^{2} + 114x – 2. |