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Transformation of Functions

Transformation of Functions. College Algebra Section 1.6. 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

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Transformation of Functions

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  1. Transformation of Functions College Algebra Section 1.6

  2. 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 Expansions and Contractions Reflections

  3. 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Recognize transformation Recognize transformation The inside part of the function The inside part of the function has been replaced by has been replaced by

  4. How to recognize a horizontal shift. Basic function Transformed function Recognize transformation The inside part of the function has been replaced by

  5. The effect of the transformation on the graph Replacing x with x – numberSHIFTS the basic graph number units to the right Replacing x with x + numberSHIFTS the basic graph number units to the left

  6. The graph of Is like the graph of SHIFTED 2 units to the right

  7. The graph of Is like the graph of SHIFTED 3 units to the left

  8. 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

  9. 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

  10. The effect of the transformation on the graph Replacing function with function – numberSHIFTS the basic graph number units down Replacing function with function + numberSHIFTS the basic graph number units up

  11. The graph of Is like the graph of SHIFTED 3 units up

  12. The graph of Is like the graph of SHIFTED 2 units down

  13. How to recognize a horizontal expansion or contraction 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

  14. How to recognize a horizontal expansion or contraction Basic function Transformed function Recognize transformation The inside part of the function Has been replaced with

  15. 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.

  16. The graph of Is like the graph of CONTRACTED 3 times

  17. The graph of Is like the graph of EXPANDED 3 times

  18. How to recognize a vertical expansion or contraction 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

  19. 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.

  20. The graph of Is like the graph of EXPANDED 3 times vertically

  21. The graph of Is like the graph of CONTRACTED 2 times vertically

  22. 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

  23. The graph of Is like the graph of FLIPPED horizontally

  24. 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 -1 FLIPS the basic graph vertically

  25. The graph of Is like the graph of FLIPPED vertically

  26. g(x) Write the equation of the given graph g(x). The original function was f(x) =x2 (a) (b) (c) (d)

  27. Example

  28. 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.

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