Transforming graphs
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Transforming Graphs. Learning Objectives: Understand how graphs of functions are transformed. Assumed knowledge. Example. Graphs:. translation. y = x 2 ± c. Plot using a graphic calculator and then sketch y=x 2 , y=x 2 +3 and y=x 2 - 2

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Transforming Graphs

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Transforming graphs

Transforming Graphs

Learning Objectives:

Understand how graphs of functions are transformed.


Assumed knowledge

Assumed knowledge


Example

Example


Graphs

Graphs:

translation

y = x2±c

  • Plot using a graphic calculator and then sketch y=x2, y=x2 +3 and y=x2 - 2

  • How is the graph of y=x2 transformed to make these two graphs?

  • For y=x2 +3 there is a translation of

[ ]

0

3

[ ]

0

-2

For y=x2 - 2 there is a translation of


Graphs translation

Graphs: translation

y = (x ±k)2

  • Plot using a graphic calculator and then sketch y=x2, y=(x- 3)2 and y=(x + 1)2

  • How is the graph of y=x2 transformed to make these two graphs?

  • For y=(x- 3)2 there is a translation of

  • For y =(x + 1)2 there is a translation of

[ ]

3

0

[ ]

-1

0


Have a go

Have a go

  • New Try These

    • What transformations are occurring?

y = x2 + 3x + 1

y = (x-3)2 + 3(x-3) + 1

y + 4 = x2 + 3x + 1

y + 4 = (x - 3)2 + 3(x - 3) + 1


Example1

Example

y = x2 + 3x + 1


Example2

[ ]

3

0

is translated by

Example

y = (x-3)2 + 3(x-3) + 1

y = x2 + 3x + 1


Example3

[ ]

0

-4

is translated by

Example

y + 4 = x2 + 3x + 1

y = x2 + 3x + 1

y = x2 + 3x + 1

y = x2 + 3x + 1 - 4


Example4

[ ]

3

-4

is translated by

Example

y = (x - 3)2 + 3(x - 3) + 1 - 4

y = x2 + 3x + 1


Translation rules in a nutshell 1

[ ]

a

0

Translation Rules -in a nutshell (1)

For any graphy = f(x)

The translation y = f(x - a)

moves it ‘a’ units to the right

The translation y = f(x + a)

moves it ‘a’ units to the left

i.e. for y = f(x - a) is translated by

Click to see


Translation rules in a nutshell 2

[ ]

0

b

Translation Rules -in a nutshell (2)

For any graphy = f(x)

The translation y = f(x) + b

moves it ‘b’ units up

… this can be considered as y - b = f(x)

i.e. for y - b = f(x) is translated by

Same as ‘y = f(x) + b’

Click to see


Translation rules in a nutshell 3

[ ]

a

b

Translation Rules -in a nutshell (3)

The previous 2 rules can be combined….

For any graphy = f(x)

i.e. for y - b = f(x - a) is translated by

Same as y = f(x - a) + b

i.e. ‘a’ units right

…. and ‘b’ up


Graphs translation1

Graphs: translation

  • Plot using a graphic calculator and then sketch y=x3, y=x3 +1 y=x3– 3

  • How is the y=x3 transformed to make the other two graphs?

  • For y=x3 +1 there is a translation of 1 unit up.

  • For y=x3– 3 there is a translation of -3 unit up.

  • What about y=x4 + 2 or y= x3 + x2


Graphs translation2

Graphs: translation

  • Plot using a graphic calculator and then sketch y=sin (x), y=sin (x) +1 y=sin (x)– 3 work in degrees.

  • How is the y=sin x transformed to make the other two graphs?

  • For y=sin (x) + 1 there is a translation of 1 unit up.

  • For y=sin (x) - 3 there is a translation of -3 unit up.

  • What about y=cos (x) + 2 or y=tan (x) – 4?


Graphs more translation

Graphs: more translation

  • Plot using a graphic calculator and then sketch y=sin x, y=sin( x+ 90) and y=sin( x - 45).

  • How is the y=sin x transformed to make these two graphs?

  • For y=sin( x+ 90) there is a translation of -90 units in the x direction.

  • For y=sin( x - 45). there is a translation of 45 units in the x direction.


Our old friend completing the square

Vertex must be at (-5,-9)

Our old friend : completing the square

Find translation from y=x2 by writing in completed square form.


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