Download Parent Functions

Warm up 8/2
For each function, evaluate f(0), f(1/2),
and f(-2)
1.f(x) = x2 – 4x
2.f(x) = -2x + 1
3.If f(x) = -3x, find f(2x) and f(x-1)
4.If f(x) = -2x + 3, find f(-2x) and f(2x-1)
Answers
1) f(0)=0, f(1/2)=-1.75, f(-2)=12
2)
=1,
=0,
=5
3) -6x , -3x + 3
4) 4x + 3, -4x + 5
Lesson 1.8 Transformations
What is a
translation?
A translation is type of
transformation where a
graph is moved
horizontally and/or
vertically.
Given the graph f(x)=(x-h)+k.


The graph moves horizontally (h)
units and vertically (k) units
So f(x) = (x-h) + k
Left/right
Opposite of h
Up/down
Example 1:
If the pre-image (original) is f(x) = 2x,
 Describe the translation of the image of
f(x) = 2(x – 3)+ 4.
3 units to the right
h = _____
which means _____________
3
4 units up
4
k = _____
which means______________
Example 2:
Pre-image f(x) = 3x
Describe the
translation.
Image f(x) = 3(x+2) - 3
left 2, down 3
Example 3:
Write the new equation.
1
The graph f ( x)  x is
3
translated 2 units left and 4 units up.
1
f ( x )  ( x  2)  4
3
Example 4: Given f(x) = -4x.

A. Find f(x+5).
-4(x+5)
-4x – 20

B. Find f(x-1)+6.
-4 (x-1)+6
-4x +4 + 6
-4x + 10
Example 5:
The pre-image is the blue
function defined as y =x
a) What would be the
equation of the red
function?
y=x+3
b) What would be the
equation of the green
function?
y = (x – 3) – 1
Another type of transformation is a REFLECTION…
Reflection across
the y-axis
Reflection across
the x-axis
Each point flips
across the y-axis
The x-coordinate
changes
(x,y)
(-x, y)
Each point flips
across the x-axis
The y-coordinate
changes
(x,y)
(x,-y)
Translating and Reflecting Functions
Use a table to perform
each
transformation of
y = f(x).
a) Translation 2 units
down
b) Reflection across
the y-axis
Stretches and Compressions
Stretch
Compression/
Shrink
Horizontal
Vertical
Each point is
pulled away from
the y-axis. The xcoordinate
changes.
(x, y)
(bx, y)
Each point is
pulled away from
the x-axis. The ycoordinate
changes.
(x, y)
(x, by)
Each point is
pushed toward
the y-axis. The xcoordinate
changes.
(x, y)
(bx, y)
Each point is
pushed toward
the x-axis. The ycoordinate
changes.
(x, y)
(x, by)
Use a table to
perform a
horizontal
compression of
y = f(x) by a
factor of ½.
Lesson 1.9 - Intro to Parent
Functions
The Parent Function is the simplest
function with the defining
characteristics of the family.
Functions in the same family are
transformations of their parent
functions.
Parent Functions
Family
constant
Linear
quadratic
Rule
f(x) = c
f(x) = x
f(x) = x2
Domain
  x  
  x  
  x  
Range
y=c
  y  
y≥0
Parent Functions
Family
Cubic
Square Root
Rule
f(x) = x3
f ( x) 
Domain
  x  
x≥0
Range
  y  
y≥0
x
Identify the parent function and
describe the transformation
1.
2.
3.
f ( x) 
x2
f ( x)  x
f(x) = x2
f(x) = x
f ( x)  x
Up 4
Down 3
Right 2
f(x) = x2 + 4
f(x) = x-3
f ( x)  x  2
Find the parent function and the
transformation
1. Graph it
x
-4
-2
0
2
4
y
8
2
0
2
8
Parent function:
f(x) = x2
2. Look at some points. Compare (2,2) with (2,4) from
the parent function.
Both x values are the same. Starting with the 4
(parent function) what did we do to = 2?
4/2 = 2
So each y value was divided by 2. That is a vertical
compression