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  • These notes were made using the textbook and “reteaching”

2.6 Presentation Transcript

  • 1. 2.6 FAMILIES OF FUNCTIONS
  • 2. FAMILIES OF FUNCTIONS
    • There are sets of functions, called families that share certain characteristics.
      • A parent function is the simplest form in a set of functions that form a family.
      • Each function in the family is a transformation of the parent function
  • 3. FAMILIES OF FUNCTIONS
  • 4. TRANSLATIONS
    • One type of transformation is a translation .
      • A translation shifts the graph of the parent function horizontally, vertically, or both without changing shape or orientation.
  • 5.  
  • 6. EXAMPLE
    • How can you represent each translation of y = |x| graphically?
    Shift the parent graph down 2 units Shift the parent graph left 1 unit
  • 7. EXAMPLE
    • How can you represent each translation of y = |x| graphically?
    Shift the parent graph right 3 units and up 1 units Shift the parent graph left 2 units and down 3 units
  • 8. REFLECTIONS
    • A reflection flips the graph over a line (such as the x – or y – axis)
      • Each point on the graph of the reflected function is the same distance from the line of reflection as its corresponding point on the graph of the original function.
  • 9. REFLECTIONS When you reflect a graph in the y-axis, the x values change signs and the y-values stay the same. When you reflect a graph in the x-axis, the y-values change signs and the x-values stay the same.
  • 10.  
  • 11. REFLECTING A FUNCTION ALGEBRAICALLY
    • Let and be the reflection in the x-axis. What is a function rule for ?
  • 12. REFLECTING A FUNCTION ALGEBRAICALLY
    • Let and be the reflection in the
    • y-axis. What is a function rule for ?
  • 13. 2.6 CONTINUED
  • 14. VERTICAL STRETCH AND VERTICAL COMPRESSION
    • A vertical stretch multiplies all y-values of a function by the same factor greater than 1.
    • A vertical compression reduces all
    • y-values of a function by the same factor between 0 and 1.
    Why do you think the value being multiplied is always positive?
  • 15.  
  • 16. EXAMPLE
    • The table represents the function f(x). Complete the table to find the vertical stretch and vertical compression. Then graph the functions.
  • 17.  
  • 18. EXAMPLE: COMBINING TRANSFORMATIONS
    • The graph of g(x) is the graph of f(x) = 4x compressed vertically by the factor ½ and then reflected in the y-axis. What is the function rule for g(x)?
  • 19. EXAMPLE: COMBINING TRANSFORMATIONS
    • The graph of g(x) is the graph of f(x) = x stretched vertically by the factor 2 and then translated down 3 units. What is the function rule for g(x)?
  • 20. EXAMPLE: COMBINING TRANSFORMATIONS
    • What transformations change the graph of f(x) to the graph of g(x)?