transformation of functions college algebra section 1.6
TRANSCRIPT
Transformation of Functions
College Algebra
Section 1.6
Horizontal and Vertical Shifts
Expansions and ContractionsReflections
Three kinds of TransformationsA function involving more than
one transformation can be graphed by performing transformations in the following order:
1.Horizontal shifting
2.Stretching or shrinking
3.Reflecting
4.Vertical shifting
How to recognize a horizontal shift.
Basic function
Transformed function
Recognize transformation
x
1x
The inside part of the function
has been replaced by
x
1x
Basic function
Transformed function
Recognize transformation
x
x 2
The inside part of the function
has been replaced by
x
x 2
How to recognize a horizontal shift.
Basic function
Transformed function
Recognize transformation
3x
35x
The inside part of the function
has been replaced by
x
5x
The effect of the transformation on the graph
Replacing x with x – number SHIFTS the basic graph number units to the right
Replacing x with x + number SHIFTS the basic graph number units to the left
The graph of f x x( ) ( ) 2f x x( ) ( ) 2 2
Is like the graph of
SHIFTED 2 units to the right
The graph of Is like the graph of f x x( ) 3 f x x( )
SHIFTED 3 units to the left
How to recognize a vertical shift.
Basic function
Transformed function
Recognize transformation
x
x 2
The inside part of the functionremains the same
2 is THEN subtracted
2Original function
Basic function
Transformed function
Recognize transformation
x
15x
The inside part of the functionremains the same
15 is THEN subtracted
15Original function
How to recognize a vertical shift.
Basic function
Transformed function
Recognize transformation
2x
2 3x
The inside part of the functionremains the same
3 is THEN added
3Original function
The effect of the transformation on the graph
Replacing function with function – number SHIFTS the basic graph number units down
Replacing function with function + number SHIFTS the basic graph number units up
The graph of Is like the graph of f x x( ) 3 f x x( )
SHIFTED 3 units up
The graph of Is like the graph of f x x( ) 3 2 f x x( ) 3
SHIFTED 2 units down
How to recognize a horizontal expansion or contraction
Basic function
Transformed function
Recognize transformation
x
2x
The inside part of the function
Has been replaced with
x
2x
Basic function
Transformed function
Recognize transformation
x
3x
The inside part of the function
Has been replaced with
x
3x
How to recognize a horizontal expansion or contraction
Basic function
Transformed function
Recognize transformation
3x
32x
The inside part of the function
Has been replaced with
x
2x
The effect of the transformation on the graph
Replacing x with number*x CONTRACTS
the basic graph horizontally if number is greater than 1.
Replacing x with number*x EXPANDS
the basic graph horizontally if number is less than 1.
The graph of Is like the graph of f x x( ) 3 f x x( )
CONTRACTED 3 times
The graph of Is like the graph of f x x( ) 13
2bg f x x( ) 2
EXPANDED 3 times
How to recognize a vertical expansion or contraction
Basic function
Transformed function
Recognize transformation
x
2 x
The inside part of the functionremains the same
2 is THEN multiplied
2 * Original function
Basic function
Transformed function
Recognize transformation
x3
4 3x
The inside part of the functionremains the same
4 is THEN multiplied
4 * Original function
The effect of the transformation on the graph
Replacing function with number*function CONTRACTS
the basic graph vertically if number is less than 1.
Replacing function with number* function EXPANDS
the basic graph vertically if number is greater than 1
The graph of Is like the graph of f x x( ) ( )3 3 f x x( ) 3
EXPANDED 3 times vertically
The graph of Is like the graph of f x x( ) 12
f x x( )
CONTRACTED 2 times vertically
How to recognize a horizontal reflection.
Basic function
Transformed function
Recognize transformation
x
The inside part of the function
has been replaced by
x x
xBasic function
Transformed function
Recognize transformation
x
The inside part of the function
has been replaced by
x x
x
The effect of the transformation on the graph
Replacing x with -x FLIPS the basic graph horizontally
The graph of Is like the graph of f x x( ) f x x( )
FLIPPED horizontally
How to recognize a vertical reflection.
Basic function
Transformed function
Recognize transformation
x
The inside part of the function remains the same
The function is then multiplied by -1
x
1* Original function
The effect of the transformation on the graph
Multiplying function by -1 FLIPS the basic graph vertically
The graph of Is like the graph of f x x( ) f x x( )
FLIPPED vertically
(a)
(b)
(c)
(d)
x
y
Write the equation of the given graph g(x). The original function was f(x) =x2
g(x)
2
2
2
2
( ) ( 4) 3
( ) ( 4) 3
( ) ( 4) 3
( ) ( 4) 3
g x x
g x x
g x x
g x x
Example
x
y
Given the graph of f(x) below, graph - ( 2) 1.f x
Summary ofGraph Transformations
• Vertical Translation: • y = f(x) + k Shift graph of y = f (x) up k units.• y = f(x) – k Shift graph of y = f (x) down k units.
• Horizontal Translation: y = f (x + h) • y = f (x + h) Shift graph of y = f (x) left h units.• y = f (x – h) Shift graph of y = f (x) right h units.
• Reflection: y = –f (x) Reflect the graph of y = f (x) over the x axis.
• Reflection: y = f (-x)
Reflect the graph of y = f(x) over the y axis. • Vertical Stretch and Shrink: y = Af (x)
• A > 1: Stretch graph of y = f (x) vertically by multiplying each ordinate value by A.
• 0 < A < 1: Shrink graph of y = f (x) vertically by multiplying each ordinate value by A.
• Horizontal Stretch and Shrink: y = Af (x)• A > 1: Shrink graph of y = f (x) horizontally by multiplying
each ordinate value by 1/A.• 0 < A < 1: Stretch graph of y = f (x) vertically by multiplying
each ordinate value by 1/A.