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Transforming Graphs

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Transforming Graphs

Learning Objectives:

Understand how graphs of functions are transformed.

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

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

- 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

y = x2 + 3x + 1

[ ]

3

0

is translated by

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

y = x2 + 3x + 1

[ ]

0

-4

is translated by

y + 4 = x2 + 3x + 1

y = x2 + 3x + 1

y = x2 + 3x + 1

y = x2 + 3x + 1 - 4

[ ]

3

-4

is translated by

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

y = x2 + 3x + 1

[ ]

a

0

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

[ ]

0

b

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

[ ]

a

b

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

- 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

- 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?

- 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.

Vertex must be at (-5,-9)

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