2.6

2.6 FAMILIES OF FUNCTIONS

FAMILIES OF FUNCTIONS

There are sets of functions, called families that share certain characteristics. A parent function is the simplest form in a set

of functions that form a family. Each function in the family is a transformation

of the parent function

FAMILIES OF FUNCTIONS

TRANSLATIONS

One type of transformation is a translation. A translation shifts the graph of the parent

function horizontally, vertically, or both without changing shape or orientation.

EXAMPLEHow can you represent each translation of y = |x| graphically?

1.

2.

2g x x

1h x x

Shift the parent graph

down 2 units

Shift the parent graph

left 1 unit

EXAMPLE

How can you represent each translation of y = |x| graphically?

1.

2.

3 1g x x

2 3h x x

Shift the parent graph right 3 units and up 1 units

Shift the parent graph left 2 units and down 3 units

REFLECTIONS

A reflection flips the graph over a line (such as the x – or y – axis) Each point on the graph of the reflected function

is the same distance from the line of reflection as its corresponding point on the graph of the original function.

REFLECTIONSWhen you reflect a graph in the y-axis, the x values change signs and the y-values stay the same.

When you reflect a graph in the x-axis, the y-values change signs and the x-values stay the same.

REFLECTING A FUNCTION ALGEBRAICALLY

Let and be the reflection in the x-axis. What is a function rule for

?

3 3f x x g x g x

g x f x

REFLECTING A FUNCTION ALGEBRAICALLY

Let and be the reflection in the y-axis. What is a function rule for ?

3 3f x x g x g x

g x f x

2.6 CONTINUED

VERTICAL STRETCH AND VERTICAL COMPRESSION A vertical stretch multiplies all y-values of

a function by the same factor greater than 1. A vertical compression reduces all

y-values of a function by the same factor between 0 and 1.

Why do you think the value being multiplied is always positive?

EXAMPLEThe table represents the function f(x). Complete the table to find the vertical stretch and vertical compression. Then graph the functions.

EXAMPLE: COMBINING TRANSFORMATIONS

The graph of g(x) is the graph of f(x) = 4x compressed vertically by the factor ½ and then reflected in the y-axis. What is the function rule for g(x)?

EXAMPLE: COMBINING TRANSFORMATIONS

The graph of g(x) is the graph of f(x) = x stretched vertically by the factor 2 and then translated down 3 units. What is the function rule for g(x)?

EXAMPLE: COMBINING TRANSFORMATIONS

What transformations change the graph of f(x) to the graph of g(x)?

2 22 6 1f x x g x x