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

Warm-up.

452 2) x

1)( 1) 2

3

x

xxf

For each function below, a) state the domain b) even/odd/neither c) symmetry

Suppose

Warm-up.

xxxxf 4)( 23

1) If , what is x?

2) Find all intercepts of the graph of f

4)( xf

Suppose and are points on a line.

Write the equation of the line containing these 2 points.

Warm-up.

3)1( f 7)2( f

Warm-up.

1. Evaluate the following:

2. State the domain for this function

3. Sketch the graph

40 if 04 if 2

4 if 1

xx x

xxf

)0()1()4()5(

ffff

2.6 Function Transformations

2.6 Function Transformations

a. Vertical Shift

f (x) x 2 2Parent function :

Shift Down 2 units

2x

Vertical Shift (or translation) shifts UP k units

shifts DOWN k units

kxf )(

kxf )(

b. Horizontal Shift

f (x) (x 3)2

Parent function : 2x

Shift left 3 units

Horizontal shift (or translation) shifts LEFT h units

shifts RIGHT h units

)( hxf

)( hxf

2a. Reflection about the x-axis

f (x) xParent function : x

Reflect over x-axis.

Reflects graph about the x-axis)(xf

2b. Reflects graph about the y-axis

f (x) xParent function :

Reflect over y-axis.

x

Reflects graph about the y-axis)( xf

3a. Stretch (dilate) the graph vertically

f (x) 2 x

)(xaf

Parent function :

Stretch vertically by : 2

|| x

If a > 1, stretches graph vertically

If 0 < a < 1, compresses graph vertically

)(xaf

3b. Horizontal Stretch/Compress

f (x) 12

x

)(bxf

Horizontal Scale

If b > 1, compresses horizontally (x-values by 1/b)If 0 < b < 1, stretches horizontally (x-values by 1/b)

)(bxf

3b. Horizontal Dilation (Scale)

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

vertical dilation

32)( xxf

Practice

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 ?

5. a) Rewrite function in standard form

Step 1: Always, factor out coefficients and write in standard form, before doing transformations!

khxbfaxf ))((

Rewrite in standard form:

23 xy

Perform the transformations in this order

khxbfa )(

1.Vertical scale Vertical shift

4.

Horizontal shift3.

Horizontal scale2.

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

22)( 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) 5)

3)

6. More Practice…

3)1(2)( 2 xxf

2)()( 3 xxf

1|3|2)( xxf

1)2()( xxf

452)( xxf

7. Domain

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

xxf )(

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

11. Write an equation from the graph

1. Identify parent (shape)

2. Compare key points to determine if y values are scaled.

3. Observe translations and reflections

4. Write in standard form khxbfaxf ))((

1. Library of Functions (Take note of key points)

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Linear Function( )f x mx b

2

Square Function

( )f x x 3

Cube Function

( )f x x

Square Root Function

( )f x x

3

Cube Root

( )f x x

Reciprocal1( )f xx

Absolute Value( )f x x

“Slope” = 1

Move:Right 1, Up 1

to next point on graph

1

1

1

1

College Algebra Notes 2.6 Write the Function from the GraphFor each graph below:a)Name the parent function b) Describe the sequence of transformations (in order) c) Determine the function that describes the graph d) Verify key points by plugging into your function.

1) 2)

3) 4)

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

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

x

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

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.

yx

byx ,1,

ayxyx ,,