**Transforming Functions**

**Definition of a Transformation**

In other words, Moving a shape/line so that it is in a different position, but still has the same size, area, angles and line lengths.

Reflections Over X and Y-Axis

Vertical and Horizontal Stretch

Vertical Stretch

graph is moving away from x-axis

a > 1

2f(x)

if negative:

take absolute value to perform stretch

reflect over the x-axis

Horizontal Stretch

graph is moving away from the y-axis

0 < a < 1

f(0.5x)

If a is negative:

take absolute value to stretch

reflect over y-axis

Parent function - green

Horizontal stretch - red

Horizontal compression - blue

Parent Function - green

Vertical stretch - blue

Vertical compression - red

Example

**Compression**

Reflections over X- Axis

The output value changes.

f(x)----> -f(x)

Reflection across Y-Axis

The input value changes

f(x)---> f(-x)

Placing the edge of a mirror on the y-axis will form a reflection in the y-axis. This can also be thought of as "folding" over the y-axis.

Left Shift- f(x)= f(x+a)

Horizontal Translation/Shift

Right Shift- f(x)= f(x-a)

X-Axis

Examples Using Reflections

Y-axis

**Vertical Shift**

Example

A transformation where the function is made smaller (compressed).

Add/subtract a number to the function to move it up and down.

Move up: f(x)+D

Move down: f(x)–D

Example

f(x)=x^2

Move right 10 units f(x−3)=(x−3)^2

Move left 10 units f(x+3)=(x+3)^2

f(x)=x^2

Move up 2 units: f(x)+2=x^2+2

Move down 2 units: f(x)−2=x^2−2

Any figure which is moved from one location to another location on the coordinate plane without changing its shape, size, or orientation is called translation.

Example:

If the original (parent) function is y = f (x), the reflection over the x-axis is function -f (x).

Example: If the original (parent) function is y = f (x), the reflection over the y-axis is function f (-x).

When the coefficient is greater than 1, the function is compressed horizontally.

C > 1, f(x) = 2x

If the coefficient is less than 1, but greater than zero, the function is vertically compressed.

0 < C < 1,

f(x) = (1/2)x