transformation of functions college algebra section 1.6
TRANSCRIPT
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Transformation of Functions
College Algebra
Section 1.6
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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
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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has been replaced by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has been replaced by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
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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has been replaced by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
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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
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The graph of f x x( ) ( ) 2f x x( ) ( ) 2 2
Is like the graph of
SHIFTED 2 units to the right
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The graph of Is like the graph of f x x( ) 3 f x x( )
SHIFTED 3 units to the left
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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
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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
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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
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The graph of Is like the graph of f x x( ) 3 f x x( )
SHIFTED 3 units up
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The graph of Is like the graph of f x x( ) 3 2 f x x( ) 3
SHIFTED 2 units down
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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
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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
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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.
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The graph of Is like the graph of f x x( ) 3 f x x( )
CONTRACTED 3 times
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The graph of Is like the graph of f x x( ) 13
2bg f x x( ) 2
EXPANDED 3 times
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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
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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
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The graph of Is like the graph of f x x( ) ( )3 3 f x x( ) 3
EXPANDED 3 times vertically
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The graph of Is like the graph of f x x( ) 12
f x x( )
CONTRACTED 2 times vertically
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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
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The graph of Is like the graph of f x x( ) f x x( )
FLIPPED horizontally
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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
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The graph of Is like the graph of f x x( ) f x x( )
FLIPPED vertically
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(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
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Example
x
y
Given the graph of f(x) below, graph - ( 2) 1.f x
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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.