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2.6 Function Transformations. 1. Transformations. To graph: Identify parent function and adjust key points. 1. Translations (Shift). Vertical Shift (or translation) shifts UP k units shifts DOWN k units . Horizontal shift (or translation) shifts LEFT h units - PowerPoint PPT Presentation
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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