What is a transformation?

A transformation changes the position or shape of a graph without changing the basic function.

Think of it like moving or stretching a picture.

The parent function

The parent function is the simplest version.

Example:

f(x) = x^2

All transformations come from modifying this base.

Types of transformations

1. Vertical Shifts

f(x)+k

• k>0: move up

• k<0: move down

Example:

x^2 + 3

Moves the graph up 3 units

2. Horizontal Shifts

f(x−h)

• h>0: move right

• h<0: move left

Example:

(x – 2)^2

Moves right 2 units

3. Vertical Stretch & Compression

a f(x)

• a>1: stretch

• 0<a<1: compress

Example:

2x^2

Makes the graph taller

4. Reflections

• Over x-axis: −f(x)

• Over y-axis: $f(-x)$

Transformation Order (important!)

Order matters:

1. Inside (horizontal)

2. Outside (vertical)