Lesson 1.6
Transformation of Functions
1.3 Additional Example 1
Graph the function and give the range.
0 if
𝑥 ≤ −3
𝑓 𝑥 = −3x if − 3 ≤ 𝑥 < 0
𝑥 2 if 𝑥 ≥ 0
1.3 Additional Example 2
Given that 𝑓 𝑥 =
3
,
𝑥
find and simplify
𝑓 𝑥 + ℎ − 𝑓(𝑥)
.
ℎ
Common Graphs
 There
are some very common graphs that
you should familiarize yourself with.
 See
p. 205 for a nice chart.
Common Graphs
Common Graphs (cont.)
Common Graphs (cont.)
The Goal
Our goal is to be able to understand the
connection between the manipulation of
the equations and their graphs.
For Example . . .
𝒇 𝒙 = 𝒙𝟐
𝒇 𝒙 = 𝒙𝟐 + 𝟐
For Example . . .
𝒇 𝒙 = 𝒙𝟐
𝒇 𝒙 = (𝒙 + 𝟑)𝟐
For Example . . .
𝒇 𝒙 = 𝒙𝟐
𝒇 𝒙 = 𝟐𝒙𝟐
For Example . . .
𝒇 𝒙 = 𝒙𝟐
𝒇 𝒙 = (𝟑𝒙)𝟐
For Example . . .
𝒇 𝒙 = 𝒙𝟐
𝒇 𝒙 = (𝒙 + 𝟑)𝟐 +𝟐
Two Tips to Remember
 Horizontal
changes always work the
opposite of how you think they should.
 Vertical
changes always work
how you think they should.
exactly
Translations
Vertical Shifts
Equation
Graphing
𝑦 = 𝑓 𝑥 + 𝑐 Shift up c units
𝑦 =𝑓 𝑥 −𝑐
Shift down c units
Horizontal Shifts
Equation
𝑦 =𝑓 𝑥+𝑐
𝑦 =𝑓 𝑥−𝑐
Graphing
Shift left c units
Shift right c units
(𝒄 > 0 below)
Example
𝑦 = 𝑥+5
𝑦 = 𝑥−5
(𝒄 > 0 below)
Example
𝑦 = 𝑥+5
𝑦 = 𝑥−5
Stretching & Shrinking
Vertical Stretching and Shrinking
(𝒄 > 0 below)
Equation
Graphing
𝑦 = 𝑐𝑓 𝑥 , 𝑐 > 1
Vertical stretch by a
factor of c
𝑦 = 𝑐𝑓 𝑥 , 0 < 𝑐 < 1
Vertical shrink by a
factor of 1/c
Horizontal Stretching and Shrinking
Example
𝑦=5 𝑥
𝑦=
1
𝑥
5
(𝒄 > 0 below)
Equation
Graphing
𝑦 = 𝑓 𝑐𝑥 , 𝑐 > 1
Horizontal shrink by a
factor of 1/c
𝑦 = 𝑓 𝑐𝑥 , 0 < 𝑐 < 1
Horizontal stretch by
a factor of c
Example
𝑦 = 5𝑥
𝑦=
1
𝑥
5
Reflections
Reflection About the x-axis
Equation
𝑦 = −𝑓 𝑥
Graphing
Flip graph over x-axis
Example
𝑦=− 𝑥
Reflection About the y-axis
Equation
𝑦 = 𝑓 −𝑥
Graphing
Flip graph over y-axis
Example
𝑦 = −𝑥
Example 1
Graph each
function.
a.
𝑔 𝑥 =𝑓 𝑥+1 −2
Example 1
Graph each
function.
b.
𝑔 𝑥 = −𝑓 𝑥 + 1
Example 1
Graph each
function.
c.
𝑔 𝑥 =𝑓
1
𝑥
2
Example 2
y
10
Graph.
𝑔 𝑥 =2 𝑥+2−2
x
-10
-10
10
Example 3
y
10
Graph.
3
𝑔 𝑥 =− 𝑥+2
x
-10
-10
10
Example 4
y
10
Graph.
1
𝑔 𝑥 = (𝑥 + 1)2
2
x
-10
-10
10
Questions???
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