Warm Up:
• Have your handout from yesterday on your desk
• We will not be going over quizzes unless there is time at the end of class
• In your notes, explain what a parent function
is and why it is important to recognize parent
functions.
Homework Q & A
Families of Functions
Objective:
I can understand transformations of
functions.
Write in your notebook ONLY what you see in the
yellow boxes [except for this yellow box ].
Everything will be done on your graphing calculator
today.
Vocabulary
Parent Function
• Simplest form in a set of functions.
Transformation:
• Change in the size or position of a function
Translation:
• Moves a function horizontally or vertically
Reflection:
• Reflects a function across a line of reflection
Dilation:
• Changes a function size
• Set your calculator
window to:  6  x  6
6  y  6
• Graph
Y1  x
2
Translations
y
6
4
2
• Graph
f (x )
Y2  x 2  3
x
-2
•
x
Graph
0
1
2
3
y1
y
Y3  x 2 2 4
0
3
1
4
4
7
9
12
f ( x)  k
-4
-6
-6
-4
-2
2
4
6
Vertical Translation: k units
Up:
k 0
Down: k  0
Translations
• Graph
• Graph
• Graph
x
0
1
2
3
4
5
y
6
Y1  x 2
Y2  ( x  3) 2
4
2
Y3  ( x  4) 2
y1
0
1
4
9
16
25
y2
9
4
1
0
1
4
x
f (x )
-2
f ( x  h)
-4
-6
-6
-4
-2
2
4
6
Horizontal Translation, h units
Left:
xh
Right: x  h
Reflections
y
6
• Graph
• Graph
• Graph
x
-2
0
-1
1
20
31
2
Y1 
( x)
Y2   ( x )
Y3 
y11
error
0
error
1
0
1.4
1
1.7
1.4
( x)
y23
1.4
0
1
-1
0
-1.4
error
-1.7
error
f (x )
4
2
0
x
-2
-4
-6
-6
-4
-2
0
2
Reflections:
Across x-axis
Across y-axis
4
6
 f (x )
f ( x)
y
Dilations
6
Y1  x
• Graph
• Graph Y2  2 x
• Graph Y3  0.5 x
x
-2
-1
0
1
2
y1
2
1
0
1
2
y2
4
2
0
2
4
4
2
x
f (x )
-2
a  f (x )
-4
-6
-6
-4
-2
2
4
6
Dilations: Vertical
stretch
compression
a 1
0  a 1
Transformation of f(x)
Translation: Vertical (k > 0)
Up k units
𝑓 𝑥 +𝑘
Down k units 𝑓 𝑥 − 𝑘
Dilation:
𝑎⋅𝑓 𝑥
Vertical by a factor of a
𝑎>1
Compression: 0 < 𝑎 < 1
Stretch:
Translation: Horizontal (k > 0)
Right h units
𝑓 𝑥−ℎ
Left h units
𝑓 𝑥+ℎ
Reflection
Across x-axis
−𝑓 𝑥
Across y-axis
𝑓 −𝑥
Combining Transformations
f ( x)  4 x
Find g(x) when f(x) is translated 3
units up.
g (x )  f (x )  3
 4x  3
Find g(x) when f(x) is stretched
by a factor of 0.5 and reflected
across the y-axis.
g (x )  0.5f (  x )
g ( x )  0.5( 4(  x ) )
Find g(x) when f(x) is translated
2 units left.
g (x )  f ( x  2)
 4( x  2)
 4x  8
g ( x )  0.5( 4 x )
g ( x )  2 x
p.104:10-33