Function transformations are math operations that cause the shape of a function's graph to change. We will be discussing how these function shapes are related to equations, and how changes in those equations effect the shape of the functions.

For example,
both of the following functions are parabolas, but by changing the
coefficient of x^{2} we change the shape of the parabola's graph:

We are going to work with the shapes of
functions as they are drawn on graph paper. We will use words
like ** stretch**,

It's important to understand that we are not talking about changing the graph paper. We are not stretching, compressing, or moving the graph paper. That's a different, although related, topic. You can find material on scaling graph paper here: Graph Paper - X, Y Scaling

The general term for describing a function's shape change is 'transform'. The three types of transforms are...

**Scale**- A
is the*scaling*or*stretching*of a shape.*compressing*

- A
**Translate**- A
is the*translation*of a shape without distorting it.*moving*

- A
**Rotate**- A
is the*rotation*of a shape about a point.*spinning*

- A

Our list of transform types could also
include a change in shape called a reflection.
In this discussion we will cover ** reflections** in the

We will not be covering rotations here. So, the transforms that we'll take a look at are scalings (including reflections) and translations.

At first, let's work with some terms that are used to describe shape changes. We will begin with these terms:

- Vertical scaling
- Horizontal scaling
- Vertical translation
- Horizontal translation

Be sure to clearly understand what 'vertical' and 'horizontal' mean:

The word ** horizontal** refers to an alignment or axis that extends to the

On a typical (x, y) graph, the ** x-axis** is

Vertical scaling refers to a stretching of a shape along a vertical direction.

The ** vertical size** of the shape...

- gets larger or smaller.
- grows or shrinks in both the upward and downward directions equally.
- grows or shrinks from the vertical center of the shape.

The ** horizontal size** of the shape does not
change.

Here is a simple polynomial function, a cubic, being presented with several different vertical scaling:

As the above animation cycles, you can imagine that the function's shape
is being ** stretched vertically**. The shape is stretched upward above the
x-axis, and the shape is stretched downward below the x-axis.

However, you ** may** imagine that the function is being compressed

So, the following animation is meant to help insure that you are imagining a vertical scaling and not a horizontal one:

Note the little gray markers in the above animation. The ones on the x-axis do not move; however, their corresponding y-axis markers are stretched further and further apart upward or downward. Many of the y-axis markers are stretched far enough to be past the edge of the graph paper.

Horizontal scaling refers to the stretching of a shape along a horizontal direction.

The ** horizontal size** of this shape...

- gets larger or smaller.
- grows or shrinks in both the leftward and rightward directions equally.
- grows or shrinks from the horizontal carter of the shape.

The ** vertical size** of the shape does not change.

Here is a trigonometric function, a sine curve, that is shown at several horizontal scalings:

Vertical translation refers to a movement of a shape upward or downward.

The ** vertical position** of the shape...

- moves upward or downward.

The ** horizontal position** of the shape does not
change.

The ** vertical and horizontal size** of the shape
do not change.

Here is a simple rational function, ** y = 1/x**, that is shown with
several vertical translations:

Horizontal translation refers to the movement of a shape to the left or to the right.

The ** horizontal position** of the shape...

- moves leftward or rightward.

The ** vertical position** of the shape does not
change.

The ** vertical and horizontal size** of the shape
do not change.

Here is a parabola, ** y = x^{2}**, undergoing several horizontal
translations:

Since we often work on an (x, y) graph,
and since the ** x-axis** is usually represented

Now, how are these transforms represented as elements within the equations that describe functions? Or, to turn the idea around, how do changes in the terms and factors of a function's equation effect the shape of its graph? It's probably best at this point to work with a certain function and to describe our transforms along with it. Let's start with a simple second degree polynomial, the parabola: