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War
m -
Up Identify the following:
IntervalsIncreasing:Decreasing:
X Intercepts:
Y Intercepts:
Relative Maximum(s):
Relative Minimum(s):
āā,ā1 , (1,ā)
(ā1,1)
ā2,0 , 0.5,0 , (1.5,0)
0,1
ā1,3
1, ā1
Domain:
Range:
End Behavior:
All Real Numbers
All Real Numbers
šš š„ ā ā, š¦ ā āšš š„ ā āā, š¦ ā āā
Fun
ctio
ns So farā¦
Most confusing Function Characteristics
Domain Interval of X values
Range Interval of Y values
Increasing Interval Interval of X values
Decreasing Interval Interval of X values
End Behavior Look at the far ends of the graph.
If itās pointing up, Y is approaching positive infinity.
If itās pointing down, Y is approaching negative infinity.
Ho
mew
ork
Qu
est
ion
s?
Fun
ctio
ns Objectives for today
Define Parent Functions and be able to associate the graph of a parent function with the correct name and function notation. (continued from yesterday)
Identify vertical and horizontal transformations from both a graph of a function and the function equation.
Introducing PARENT FUNCTIONS!P
are
nt
Fun
ctio
ns
Parent functions are the simplest form of families of functions.
Function Parent Function
š š„ = 2š„2 + 4 š š„ = š„2
š š„ = š„ ā 7 š š„ = š„
š š„ =1
3(š„ ā 7)3ā1 š š„ = š„3
š š„ = |š„ + 4| š š„ = |x|
Constant, f(x) = C
Domain Range
End Behavior
šš š„ ā āā, š¦ ā šš š„ ā ā, š¦ ā
Critical Points
Vertex X intercepts Y intercepts
Par
en
t Fu
nct
ion
s
Domain Range
End Behavior
šš š„ ā āā, š¦ ā šš š„ ā ā, š¦ ā
Critical Points
Vertex X intercepts Y intercepts
Par
en
t Fu
nct
ion
s
Linear, f(x)=x
Domain Range
End Behavior
šš š„ ā āā, š¦ ā šš š„ ā ā, š¦ ā
Critical Points
Vertex X intercepts Y intercepts
Par
en
t Fu
nct
ion
s
Quadratic, f(x)=x2
Domain Range
End Behavior
šš š„ ā āā, š¦ ā šš š„ ā ā, š¦ ā
Critical Points
Vertex X intercepts Y intercepts
Par
en
t Fu
nct
ion
s
Radical (Square Root), f(x)=
Par
en
t Fu
nct
ion
s
Work with a partner to complete the next five parent functions.
If youāre feeling confident complete the last function, Rational.
Weāll do that one together as a class.
Cubic, f(x)=x3
Cube Root, f(x)=
Exponential, f(x)=bx, b>1
Log, f(x)= logb(x), b>1
Absolute Value, f(x)=|x|
Par
en
t Fu
nct
ion
s
Rational, Inverse, Reciprocal, f(x)=
Domain Range
End Behavior
šš š„ ā āā, š¦ ā šš š„ ā ā, š¦ ā
Critical Points
Vertex X intercepts Y intercepts
Par
en
t Fu
nct
ion
s
Whatās different about this graph?
When a function is shifted in any way from its parent function, it is said to be transformed. We call this a transformation of a function. Functions are typically transformed either verticallyor horizontally.
Tran
sfo
rmat
ion
s
Two categories of Function Transformations
1. Rigid Transformations
The basic shape of the graph is unchanged.
Vertical ShiftsHorizontal ShiftsReflections
2. NonRigid Transformations
Cause a distortion, a change in the graph.
StretchesShrinks (Compressions)
Tran
sfo
rmat
ion
s
Parent FunctionTransformed Function Transformed Function
Quadraticf(x)=x2 Shifted
Left 3 unitsUp 2 units
Shifted Right 2 unitsDown 2 units
Some simple transformationsā¦Tr
ansf
orm
atio
ns
Parent FunctionTransformed Function Transformed Function
Cubicf(x)=x3 Shifted
Down 1 unitShifted
Right 2 unitsUp 3 units
Identify the parent function and the transformations represented in the graphs.Tr
ansf
orm
atio
ns
So how do we represent these transformations algebraically?
Today we will focus on Rigid Transformations
Tran
sfo
rmat
ion
s
Vertical Transformations
When functions are transformed on the outside of the f(x) part, you move the function up and down.
Tran
sfo
rmat
ion
s
Function Notation Description of Transformation
g š„ = š š„ Ā± š Vertical shift up C units if C is positive
Vertical shift down C units if C is negative
Vertical Transformations
Function Notation Description of Transformation
g š„ = š š„ Ā± š Vertical shift up C units if C is positive
Vertical shift down C units if C is negative
Tran
sfo
rmat
ion
s
How do we interpret this function notation?
Let š š„ = š„2 and š = 3 then š š„ = š„2 + 3
Let š š„ = š„ and š = ā4 then š š„ = š„ ā 4
Let š š„ = 2š„ and š = 7 then š š„ = 2š„ + 7
X f(x)X2
g(x)X2+3
3 9 12
2 4 7
1 1 4
0 0 3
-1 1 4
-2 4 7
-3 9 12
š š = šš + š
Tran
sfo
rmat
ion
s Letās play āWhatās going to happen to the parent function?ā
X f(x)X3
g(x)X3-1
3 27 26
2 8 7
1 1 0
0 0 -1
-1 -1 -2
-2 -8 -9
-3 -27 -28
š š = šš ā š
Tran
sfo
rmat
ion
s Letās play āWhatās going to happen to the parent function?ā
Write the equation for the transformed function represented in this graph.
š š = šš ā šTr
ansf
orm
atio
ns
Parent Function?
Critical point that can help us?
Which way did it go?
By how much?
Quadratic, š š = šš
Vertex
Down
1 unit
Write the equation for the transformed function represented in this graph.
š š = šš ā šTr
ansf
orm
atio
ns
Parent Function?
Critical point that can help us?
Which way did it go?
By how much?
Log, š š = šš
Intercepts
Down
2 units
Write the equation for the transformed function represented in this graph.
š š = š + šTr
ansf
orm
atio
ns
Parent Function?
Critical point that can help us?
Which way did it go?
By how much?
Radical, š š = š
Intercepts
Up
2 units
Horizontal Translations
When functions are transformed on the inside of the āf(x) partā, you move the function left and right. Notice the direction is the oppositeof the sign inside the āf(x) partā.
Function Notation Description of Transformation
š š„ = š š„ Ā± š Horizontal shift left C units if C is positive.
Horizontal shift right C units if C is negative
Tran
sfo
rmat
ion
s
Horizontal Translations
Function Notation Description of Transformation
š š„ = š š„ Ā± š Horizontal shift left C units if C is positive.
Horizontal shift right C units if C is negative
Tran
sfo
rmat
ion
s
How do we interpret this function notation?
Let š š„ = š„2 and š = 3 then š š„ = (š„ + 3)2
Let š š„ = š„ and š = ā4 then š š„ = š„ ā 4
Let š š„ = 2š„ and š = 7 then š š„ = 2š„+7
š š = (š ā š)š
Tran
sfo
rmat
ion
s Letās play āWhatās going to happen to the parent function?ā
š š = (š + š)š
Tran
sfo
rmat
ion
s Letās play āWhatās going to happen to the parent function?ā
Write the equation for the transformed function represented in this graph.
š š = (š ā š)šTr
ansf
orm
atio
ns
Parent Function?
Critical point that can help us?
Which way did it go?
By how much?
Cubic, š š = šš
Intercepts
Left
1 unit
Write the equation for the transformed function represented in this graph.
š š = ššš(š ā š)Tr
ansf
orm
atio
ns
Parent Function?
Critical point that can help us?
Which way did it go?
By how much?
Log, š š = š³šš š
Intercepts
Right
2 units
Write the equation for the transformed function represented in this graph.
š š = (š ā š)š+šTr
ansf
orm
atio
ns
Parent Function?
Critical point that can help us?
Which way did it go?
By how much?
Cubic, š š = šš
Intercepts
Left and up
Left 2 and up 1
Reflections
When a negative sign is found on the outside of the āf(x) partā the function is flipped over the x-axis.
When a negative sign is found on the inside of the āf(x) partā the function is flipped over the y-axis.
Function Notation Description of Transformation
g š„ = āš š„ Reflected over the x-axis
g š„ = š āš„ Reflected over the y-axis
Tran
sfo
rmat
ion
s
Reflections
Function Notation Description of Transformation
g š„ = āš š„ Reflected over the x-axis
g š„ = š āš„ Reflected over the y-axis
Tran
sfo
rmat
ion
s
How do we interpret this function notation?
Let š š„ = š„2, then āš š„ = āš„2 and š āš„ = (āš„)2
Let š š„ = š„ , then ā š š„ = ā š„ and š āš„ = āš„
Let š š„ = 2š„, then āš š„ = ā2š„ and š āš„ = 2āš„
X X2 -X2
3 9 -9
2 4 -4
1 1 -1
0 0 0
-1 1 -1
-2 4 -4
-3 9 -9
š š = āšš
Reflection across the x axisTr
ansf
orm
atio
ns
X -X (-X)3
3 -3 -27
2 -2 -8
1 -1 -1
0 0 0
-1 1 1
-2 2 8
-3 3 27
š š = (āš)š
Reflection across the y axisTr
ansf
orm
atio
ns
Write the equation for the transformed function represented in this graph.
š š = ā šTr
ansf
orm
atio
ns
Parent Function?
Critical point that can help us?
Which way did it go?
Which axis has it flipped over?
Radical, š š = š
Intercepts
No Change
X-axis
Write two equations that could represent the function presented in this graph.
š š = |š|
š š = | ā š|
Tran
sfo
rmat
ion
s
Function Notation Description of Transformation
š š„ = š š„ + š¶ Vertical shift up C units
š š„ = š š„ ā š¶ Vertical shift down C units
š š„ = š š„ + š¶ Horizontal shift left C units
š š„ = š š„ ā š¶ Horizontal shift right C units
š š„ = āš(š„) Flip over the x-axis
š š„ = š(āš„) Flip over the y-axis
Summary of the Rigid Transformations
ORDER OF OPERATIONS
P Please Parentheses
E Excuse Exponents
M My Multiplication
D Dear Division
A Aunt Addition
S Sally Subtraction
Fun
ctio
ns Did we meet our objectives?
Complete the exit ticket and bring it to me to check.