Transformation of Functions
College Algebra
Section 1.6
Three kinds of Transformations
Horizontal and Vertical Shifts
Expansions and Contractions
Reflections
A 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
Basic function
x
x
Transformed function
Transformed function
x 1
x2
Recognize transformation
Recognize transformation
The inside part of the function
x
The inside part of the function
x
has been replaced by
has been replaced by
x 1
x2
How to recognize a horizontal shift.
Basic function
x
3
Transformed function
x 5
3
Recognize transformation
The inside part of the function
x
has been replaced by
x5
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 2)
2
Is like the graph of
SHIFTED 2 units to the right
f (x) (x)
2
The graph of
f ( x ) x 3 Is like the graph of f ( x )
SHIFTED 3 units to the left
x
How to recognize a vertical shift.
Basic function
x
Basic function
x
Transformed function
Transformed function
x 2
x 15
Recognize transformation
Recognize transformation
The inside part of the function
remains the same
The inside part of the function
remains the same
2 is THEN subtracted
15 is THEN subtracted
Original function 2
Original function 15
How to recognize a vertical shift.
Basic function
x
2
Transformed function
x2 3
Recognize transformation
The inside part of the function
remains the same
3 is THEN added
Original function 3
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
f (x) x 3
Is like the graph of
SHIFTED 3 units up
f (x) x
The graph of
f ( x) x 2
3
Is like the graph of
SHIFTED 2 units down
f ( x) x
3
How to recognize a horizontal expansion or contraction
Basic function
Basic function
x
x
Transformed function
Transformed function
3x
2x
Recognize transformation
Recognize transformation
The inside part of the function
The inside part of the function
x
x
Has been replaced with
Has been replaced with
3x
2x
How to recognize a horizontal expansion or contraction
Basic function
x
3
Transformed function
2x
3
Recognize transformation
The inside part of the function
x
Has been replaced with
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
f ( x ) 3x
Is like the graph of
CONTRACTED 3 times
f (x) x
The graph of
1 2
𝑓 𝑥 = 𝑥
3
Is like the graph of
EXPANDED 3 times
f ( x) x
2
How to recognize a vertical expansion or contraction
Basic function
Basic function
x
x3
Transformed function
Transformed function
2x
4x3
Recognize transformation
Recognize transformation
The inside part of the function
remains the same
The inside part of the function
remains the same
2 is THEN multiplied
4 is THEN multiplied
2 * Original function
4 * Original function
The effect of the transformation on the graph
Replacing function with number* function
EXPANDS
the basic graph vertically if number is greater than 1
Replacing function with number*function
CONTRACTS
the basic graph vertically if number is less than 1.
The graph of
f ( x ) 3( x )
3
Is like the graph of
EXPANDED 3 times vertically
f ( x) x
3
The graph of
f ( x ) 21 x
Is like the graph of
CONTRACTED 2 times vertically
f ( x) x
How to recognize a horizontal reflection.
Basic function
Basic function
x
x
Transformed function
Transformed function
x
x
Recognize transformation
Recognize transformation
The inside part of the function
The inside part of the function
x
has been replaced by x
x
has been replaced by x
The effect of the transformation on the graph
Replacing x with
graph horizontally
-x FLIPS the basic
The graph of
f ( x) x
Is like the graph of
FLIPPED horizontally
f ( x) x
How to recognize a vertical reflection.
Basic function
x
Transformed function
x
Recognize transformation
The inside part of the function remains the same
The function is then multiplied by -1
1* Original function
The effect of the transformation on the graph
Multiplying function
basic graph vertically
by
-1 FLIPS the
The graph of
f ( x) x
Is like the graph of
FLIPPED vertically
f ( x) x
g(x)
x
Write the equation of the given graph g(x).
The original function was f(x) =x2
(a) g ( x ) ( x 4) 3
2
2
g
(
x
)
(
x
4)
3
(b)
2
g
(
x
)
(
x
4)
3
(c)
(d) g ( x ) ( x 4) 3
2
Example
Given the graph of f(x) below, graph - f ( x 2) 1.
y
x
Summary of
Graph 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.
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