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Transformation of Functions
College Algebra Section 2.7
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Three kinds of Transformations
Horizontal and Vertical Shifts A function involving more than one transformation can be graphed by performing transformations in the following order: Horizontal shifting Stretching or shrinking Reflecting Vertical shifting Stretching or Shrinking Reflections
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How to recognize a horizontal shift.
Basic function Basic function Transformed function Transformed function Recognize transformation Recognize transformation The inside part of the function The inside part of the function has been replaced by has been replaced by
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How to recognize a horizontal shift.
Basic function Transformed function Recognize transformation The inside part of the function has been replaced by
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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 Is like the graph of SHIFTED 2 units to the right
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The graph of Is like the graph of SHIFTED 3 units to the left
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How to recognize a vertical shift.
Basic function Basic function Transformed function Transformed function 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 Original function
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How to recognize a vertical shift.
Basic function Transformed function Recognize transformation The inside part of the function remains the same 3 is THEN added Original 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 SHIFTED 3 units up
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The graph of Is like the graph of SHIFTED 2 units down
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How to recognize a horizontal stretch or compression
Basic function Basic function Transformed function Transformed function Recognize transformation Recognize transformation The inside part of the function The inside part of the function Has been replaced with Has been replaced with
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How to recognize a horizontal stretch or compression
Basic function Transformed function Recognize transformation The inside part of the function Has been replaced with
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The effect of the transformation on the graph
Replacing x with number*x COMPRESSES the basic graph horizontally if number is greater than 1. Replacing x with number*x STRETCHES the basic graph horizontally if number is less than 1.
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The graph of Is like the graph of Horizontally COMPRESSED 3 times
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The graph of Is like the graph of Horizontally STRETCHED 3 times
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How to recognize a vertical stretch or compression
Basic function Basic function Transformed function Transformed function 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 Original function Original function
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The effect of the transformation on the graph
Replacing function with number* function STRETCHES the basic graph vertically if number is greater than 1 Replacing function with number*function COMPRESSES the basic graph vertically if number is less than 1.
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The graph of Is like the graph of STRETCHED 3 times vertically
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The graph of Is like the graph of COMPRESSED 2 times vertically
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How to recognize a horizontal reflection.
Basic function Basic function Transformed function Transformed function Recognize transformation Recognize transformation The inside part of the function The inside part of the function has been replaced by has been replaced by 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 FLIPPED horizontally
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How to recognize a vertical reflection.
Basic function Transformed function Recognize transformation The inside part of the function remains the same The function is then multiplied by -1 Original function The effect of the transformation on the graph Multiplying function by FLIPS the basic graph vertically
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The graph of Is like the graph of FLIPPED vertically
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g(x) Write the equation of the given graph g(x). The original function was f(x) =x2 (a) (b) (c) (d)
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Example
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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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