### Transformations of the Parent Functions

```What is a Parent Function
A parent function is the
most basic version of an
algebraic function.
Types of Parent Functions
Linear
f(x) = mx + b
f(x) = x2
Square Root
f(x) = √x
Exponential
f(x) = bx
Rational
f(x) = 1/x
Logarithmic
f(x) = logbx
Absolute Value f(x) = |x|
Types of Transformations
S
V
e
r
t
i
c
a
l
T
r
a
n
s
l
a
t
i
o
n
s
V
e
r
t
i
c
a
l
c
t
h
r
e
t
VC
e o
r m
t p
i r
c e
a s
l s
i
o
n
R
e
f
l
e
c
t
i
o
n
s
O
v
e x
r a
t x
h i
e s
….More Transformations
Horizontal Translations
Horizontal S t r e t c h
Horizontal Compression
Reflections
Over the y-axis
FAMILIES TRAVEL
TOGETHER……
Families of Functions
If a, h, and k are real numbers with a≠0, then the
graph of y = a f(x–h)+k is a transformation of the
graph of y = f ( x).
All of the transformations of a function form a family
of functions.
F(x) = (a - h)+ k – Transformations should be applied
from the “inside – out” order.
Horizontal Translations
If h > 0, then the graph of y = f (x – h) is a translation of h
units to the RIGHT of the graph of the parent function.
Example: f(x) = ( x – 3)
If h<0,then the graph of y=f(x–h) is a translation of |h| units
to the LEFT of the graph of parent function.
Example: f(x) = (x + 4)
*Remember the actual transformation is (x-h), and
subtracting a negative is the same as addition.
Vertical Translations
If k>0, then the graph of y=f(x)+k is a translation
of k units UP of the graph of y = f (x).
Example: f(x) = x2 + 3
If k<0, then the graph of y=f(x)+k is a translation
of |k| units DOWN of the graph of y = f ( x).
Example: f(x) = x2 - 4
Vertical Stretch or
Compression
The graph of y = a f( x) is obtained from the graph of
the parent function by:
stretching the graph of y = f ( x) by a when a > 1.
Example: f(x) = 3x2
compressing the graph of y=f(x) by a when 0<a<1.
Example: f(x) = 1/2x2
Reflections
The graph of y = -a f(x) is reflected over the yaxis.
The graph of y = f(-x) is reflected over the x-axis.
Transformations Summarized
Y = a f( x-h) + k
V l S t o c s
e
r o Horizontal
i
r
t c
m S
otretch
or
t
p n
compression
i
r h
r
c
e
a
s
e
Horizontal
Translation
V c T l n
e a r a
r l a t
n i
t
s o
i
Multiple Transformations
Graph a function involving more than one
transformation in the following order:
Horizontal translation
Stretching or compressing
Reflecting
Vertical translation
Are we there yet?
Parent Functions
Function Families
Transformations
Multiple Transformations
Inverses
Asymptotes
Where do we go from here?
Inverses of functions
Inverse functions are reflected over the y = x line.
When given a table of values, interchange the x
and y values to find the coordinates of an inverse
function.
When given an equation, interchange the x and y
variables, and solve for y.
Asymptotes
Boundary line that a graph will not cross.
Vertical Asymptotes
Horizontal Asymptotes
Asymptotes adjust with the transformations of
the parent functions.
```