Transcript 1.6 Transformation of Functions

Unit 3
Functions (Linear and Exponentials)
Parent Functions and
Transformations
Transformation of Functions
 Recognize graphs of common functions
 Use shifts to graph functions
 Use reflections to graph functions
 Graph functions w/ sequence of transformations
The following basic graphs will be used
extensively in this section. It is important
to be able to sketch these from memory.
The identity function
f(x) = x
The exponential function
f ( x)  b x
The quadratic function
f ( x)  x
2
The square root function
f ( x)  x
The absolute value function
f ( x)  x
The cubic function
f ( x)  x
3
The rational function
1
f ( x) 
x
Transformations happen in 3
forms:
(1) translations
(2) reflections
(3) stretching.
The values that translate the graph of a
function will occur as a number added or
subtracted either inside or outside a
function.
y  f ( x  h)  k
Numbers added or subtracted inside
translate left or right, while numbers
added or subtracted outside translate up
or down.
Vertical Translation
OUTSIDE IS TRUE!
Vertical Translation
the graph of y = f(x) + k is
the graph of y = f(x) shifted
up k units;
f ( x)  x
2
f ( x)  x 2  3
the graph of y = f(x)  k is the
graph of y = f(x) shifted down
k units.
f ( x)  x 2  2
Horizontal Translation
INSIDE LIES!
Horizontal Translation
the graph of y = f(x  h) is the
graph of y = f(x) shifted right
h units;
the graph of y = f(x + h) is
the graph of y = f(x) shifted
left h units.
y   x  2
f ( x)  x 2
2
y   x  2
2
Recognizing the shift from the
equation, examples of shifting the
2
function f(x) = x
 Vertical shift of 3 units up
f ( x)  x , h( x)  x  3
2
2
 Horizontal shift of 3 units left (HINT: x’s go the opposite
direction that you might believe.)
f ( x)  x , g ( x)  ( x  3)
2
2
Example
 Explain the difference in the graphs
y  ( x  3)
y  x 3
2
2
Horizontal Shift Left 3 Units
Vertical Shift Up 3 Units
Use the basic graph to sketch the
following:
fff(f((x(xxx))))
(xxx
32)
35

2
3
Combining a vertical & horizontal shift
 Example of function that is
shifted down 4 units and
right 6 units from the
original function.
f ( x)  x
Use the basic graph to sketch the
following:
2
fff (((x
xx)))

x
xxx
 
Reflection about the x-Axis
 The graph of y = - f (x) is the graph of y = f (x)
reflected about the x-axis.
Reflection about the y-Axis
• The graph of y = f (-x) is the graph of y = f (x) reflected
about the y-axis.
Stretching and Shrinking
Graphs
Let f be a function and c a positive real number.
•If a > 1, the graph of y = a f (x) is y = f (x) vertically
stretched by multiplying each of its y-coordinates by a.
•If 0 < a < 1, the graph of y = a f (x) is y = f (x) vertically
shrunk by multiplying each of its y-coordinates by a.
g(x) = 2x2
f (x) = x2
10
9
8
7
6
5
4
3
2
1
-4 -3 -2 -1
h(x) =1/2x2
1
2
3
4
The big picture…
Sequence of Transformations
A function involving more than one transformation can be
graphed by performing transformations in the following
order.
1. Horizontal shifting
(Parentheses)
2. Vertical stretching or shrinking (Multiply)
3. Reflecting
(Multiply)
4. Vertical shifting
(Add/Subtract)
Example
 Use the graph of f(x) = x3 to graph g(x) = -2(x+3)3 - 4
10
8
6
4
2
-10 -8
-6
-4
-2
2
-2
-4
-6
-8
-10
4
6
8
10
A combination
 If the parent function is
yx
2
 Describe the graph of

y  ( x  3)  6
2
The parent would be
horizontally shifted right 3
units and vertically shifted
up 6 units
 If the parent function is
y What
 xdo we know about
3

y  2x  5
3
The graph would be
vertically shifted down 5
units and vertically
stretched two times as
much.
What can we tell about this graph?
y  (2 x)
3
It would be a cubic function reflected
across the x-axis and horizontally
compressed by a factor of ½.
Transformations of Exponential
Functions
Transformations of Graphs of Exponential
Functions
x
f
(
x
)

2
Describe the transformation(s) that the graph of
must undergo in order to obtain the graph of each of the
following functions.
State the domain, range and the horizontal asymptote for each.
f ( x)  2 x  5
29
Transformations of Graphs of Exponential
Functions
x
f
(
x
)

2
Describe the transformation(s) that the graph of
must undergo in order to obtain the graph of each of the
following functions.
State the domain, range and the horizontal asymptote for each.
f ( x)  2 x  4
30
Transformations of Graphs of Exponential
Functions
Describe the transformation(s) that the graph of f ( x)  2 x
must undergo in order to obtain the graph of each of the
following functions.
State the domain, range and the horizontal asymptote for each.
f ( x)  2 x3
31
Graph using transformations and determine the domain,
range and horizontal asymptote.
A)
B)
C)
D)