What is Shifting and Scaling in mathmatic graphs?

Shifting and Scaling can apply on most of the functions and translate them to a new graph without loosing the properties of the old graph. The follwoing are some of common functions:

- Constant Function:
`y=c` - Linear Function:
`y=x` - Quadratic Function:
`y=x^2` - Cubic Function:
`y=x^3` - Absolute Value Function:
`y=|x|` - Square Root Function:
`y=sqrt(x)`

The basic graph have the function: **y = f(x)**

All of the translations can be expressed in the form:

y = a * f [ b (x-c) ] + d

Vertical |
Horizontal | |

Scale |
a | b |

Shift |
d | c |

acts normally | acts inversely |

*To understand these graph translations, let's see what happens in simple steps with just one change at a time. *

A shift is a rigid translation in that it does not change the shape or size of the graph of the function. All that a shift will do is change the location of the graph. A vertical shift adds/subtracts a constant to/from every y-coordinate while leaving the x-coordinate unchanged. A horizontal shift adds/subtracts a constant to/from every x-coordinate while leaving the y-coordinate unchanged. Vertical and horizontal shifts can be combined into one expression.

Shifts are added/subtracted to the x or f(x) components. If the constant is grouped with the x, then it is a horizontal shift, otherwise it is a vertical shift.

**Example:**

*Vertical shift*

**y = f(x)****y = f(x)+1****1** is added to all **y** values to shift up.

Vice versa to shift down.

*Horizontal shift*

**y = f(x)****y = f(x+1)**

The height at **x** on the new graph

is equal to the height at **x+1** on the old one.

The new graph is obtained by shifting

the old one **1**
to the left. Vice versa to shift right.

A scale is a non-rigid translation in that it does alter the shape and size of the graph of the function. A scale will multiply/divide coordinates and this will change the appearance as well as the location. A vertical scaling multiplies/divides every y-coordinate by a constant while leaving the x-coordinate unchanged. A horizontal scaling multiplies/divides every x-coordinate by a constant while leaving the y-coordinate unchanged. The vertical and horizontal scalings can be combined into one expression.

Example:

*Vertical scale*

**y = f(x)****y = 2f(x)**

All **y**-values on the graph are scaled by **2**.

In subsituting **cf** for **f**,

vertical distances are scaled by **c**.

*Horizontal scale*

**y = f(x)****y = f(2x)**

The height at **x**
on the new graph is

equal to the height at **2x** on the old one.

The new graph is obtained by compressing

the old one horizontally by **2**.

In substituting **cx** for **x**,

horizontal distances are scaled by **1/c**.

**y = f(x)****y = f(2x-1)+1**