2.6 Function Transformations

2.6 Function Transformations

1. TransformationsTo graph: Identify parent function and adjust key points.Function To Graph: Move key point (x,y) to:

Vertical Shift upVertical Shift down

Horizontal Shift leftHorizontal Shift right

Reflection about x-axisReflection about y-axis

Vertical stretch if Vertical shrink if

Horizontal stretch if 0 < b <1Horizontal shrink if b > 1

cxfcxf

)()(

)()(cxfcxf

),(),(),(),(ycxyxycxyx

),(),(),(),(cyxyxcyxyx

)()(xfxf

),(),(),(),(yxyxyxyx

)(xaf ),(),( ayxyx

),1(),( yxb

yx )(bxf

1a10 a

Vertical Shift (or translation) shifts UP k units

shifts DOWN k units

1. Translations (Shift)

kxf )(

kxf )(

Horizontal shift (or translation) shifts LEFT h units

shifts RIGHT h units

)( hxf

)( hxf

a. Vertical Shift

f (x) x 2 2Parent function :

Shift Down 2 units

2x

Parent Function:

New Function

(0,0)

(1,1)

(2,4)

22 xy2xy

b. Horizontal Shift

f (x) (x 3)2

Parent function : 2x

Shift left 3 units

2. Reflections

Reflects graph about the x-axis)(xf

Reflects graph about the y-axis)( xf

2a. Reflection about the x-axis

f (x) xParent function : x

Reflect over x-axis.

2b. Reflects graph about the y-axis

f (x) xParent function :

Reflect over y-axis.

x

3. Vertical Dilation (Scale)

If a > 1, stretches graph vertically

If 0 < a < 1, compresses graph vertically

)(xaf

f (x) 2 x

3a. Stretch (dilate) the graph vertically

f (x) 2 x

)(xaf

Parent function :

Stretch vertically by : 2

|| x

3b. Horizontal Dilation (Scale)Horizontal Scale

If b > 1, compresses graph horizontallyIf 0 < b < 1, stretches graph horizontally

)(bxf

When the scale is “inside” the parent function,it is preferable to pull it OUTSIDE the parent function and apply

vertical dilation

32)( xxf

3b. Horizontal Stretch/Compress

f (x) 12x

)(bxf

4. Practice with single Transformations

Practice: p. 127 A - L

Make a table, describing the parent and transformations applied

Function Parent Transformations to apply

A) y = x2 + 2

B)

C)

D)

E)

F)

G

H)

I

J)

K)

L)

Practice

p. 127 #30

a) Graph the transformations as described

b) Write what you think the equation will be from the

description

c) graph your equation on the calculator to check your result.

Did it work out like expected?

4. Sequence of TransformationsWhen a function has multiple transformatinos applied, does

the order of the transformations matter?

23 xxf Which operation is first: Reflection or Shift ?

2)3( xxfHow about this one? Does the order matter.

5. a) Rewrite function in standard form

Step 1: Factor out coefficients

khxbfaxf ))((

When a function is written in the standard form,

Perform operations from left to right!

Examples

222)( xxf

23 xy

6. Describe sequence of Transformations

23 xyStandard Form:Parent FunctionReflection over x-axisReflection over y-axisScale yScale xShift L/RShift U/D

6. Describe sequence of TransformationsStandard Form:Parent FunctionReflection over x-axisReflection over y-axisScale yScale xShift L/RShift U/D

222)( xxf

f (x) (x 1)3 2

6. Describe sequence of Transformations

Standard Form:Parent FunctionReflection over x-axisReflection over y-axisScale yScale xShift L/RShift U/D

For each function, describe (in order) the sequence of transformations and sketch the final graph.1) 4)

2)

3)

6. More Practice…

3)2(21)( 2 xxf

2)()( 3 xxf

1|3|2)( xxf

1)2()( xxf

7. Domain

How is the domain of a function affected by the transformations?

xxf )(

2)( xxf 1)( xxf xxf )(xxf )(

Method 2: Less Preferred method

When a function is not in the standard form, perform transformations in this order:

1) Horizontal shift2) Stretch/shrink3) Reflect4) Vertical stretch Shrink

8. A second method for sequence of transformations

11. Write an equation from the graph

1. Identify parent function (look at shape)

2. Compare key points of parent function with your graph to

determine if y values are scaled.

3. Observe translations and reflections and adjust equation

accordingly.

11. Write an equation from the graph

f (x) (x 2)3

f (x) x 2 3

f (x) x 1

f (x) 2 x 3 2

Perform the transformations in this order

khxbfa )(

1.Vertical scale by a If a is negative, reflects across x-axis

Vertical shift+k: shift up k

-k : shift down k

4.

Horizontal shift-h : shift to right+h : shift to left

3.Horizontal scale by

If b is negative, reflects across y-axis

b/12.

yxb

yx ,1,

ayxyx ,,

Transformations

f (x) 1

( x) 2

1)

2)

3)

Even or Odd ?

Warm-up.a) List the sequence of transformations and sketchb) List the transformations that are made to each key point of

the parent function.

452 2) x

6121)( 1)

2

xxg

1)( 3) 2

3

xxxf