A transformation is an algebraic change to afunction which affects the shape or position of the graph.

Introduction to Transformations

What is the difference between and ?

y = f ( x) + a y = f ( x + a)

x f (x) + a y x f (x)+ a y

y = f ( x) + a y = f ( x + a)

Making a change after

the function will affect

only the

Making a change

before the function will

affect only they x-coordinate. -coordinate.

Higher Maths 1 2 2 Transformations

1

Any change to the position of a graph is called a translation.

Translation

y = f ( x) + a

Slides thegraph of vertically.

f ( x)

a

f ( x) + a

f ( x)

y = f ( x + a)

Slides the graph of horizontally.

f ( x)

a

f ( x + a) f ( x)

up

down

+ a

– aleft

right

+ a– a

Higher Maths 1 2 2 Transformations

2

If the positive and negative coordinates of a graph are

inverted, the graph will be reflected across either the x

or y-axis.

Reflection

y = - f ( x)

Reflects the graph of f

( x)

vertically across the x-axis.

- f ( x)

f ( x)

y = f (- x)

Reflects the graph of f ( x)

horizontally across the y-axis.f (- x) f ( x)

Higher Maths 1 2 2 Transformations

3

If every coordinate is multiplied by the same value, the overall shape of the graph will be distorted.

Distortion

y = a f ( x)

f ( x)

a f ( x)

Changes the vertical size of the graph of f

( x).

a > 1 ‘stretch’

a < 1 ‘compress’

y = f (a x)

f ( x)

Changes the horizontal size of the graph of f ( x).

a < 1 ‘stretch’

a > 1 ‘compress’

f (a x)

Higher Maths 1 2 2 Transformations

4

The diagram below shows the graph of . Sketch .

Sketching Composite Transformations

Higher Maths 1 2 2 Transformations

5

Example

y = -2 f ( x + 3)y = f (x)

y = f ( x)

y = f ( x + 3) y = f ( x + 3)

(3,0)

(-2,-2)

(-5,4)

(0,-2)

Remember to label all relevant coordinates.

Important

y = -2 f ( x + 3)