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4.6 – Transformation of Parent Functions

Objectives

  • Given the graph, equation, or both graph and equation of a parent function, determine the equation of the function that results from a horizontal or vertical shift to the parent function.
  • Given the graph, equation, or both graph and equation of a parent function, determine the equation of the function that results from a vertical stretch or compression to the parent function.
  • Given the graph, equation, or both graph and equation of a parent function, determine the equation of the function that results from flipping the parent function’s graph over the x-axis.
  • Given the graph, equation, or both graph and equation of a parent function, determine the equation of the function that results from a combination of shifts, stretches, or compressions to the parent function.


Key Terms

  • Transformation – A function that has been changed by shifting, stretching, compressing or flipping the original function.

 

Notes

Transformations of Functions
  • Shift: to MOVE a function vertically (up or down) or horizontally (left or right)
    • Up / Down: Add or subtract a value to the end of the equation
    • Left / Right: Subtract a positive or negative value from the x-term
  • Stretch: to change the SHAPE of a function (make it tall and skinny)
    • Multiply the x-term by a number greater than 1
  • Compress: to change the SHAPE of a function (to make it short and wide)
    • Multiply the x-term by a fraction or decimal between 0 and 1
  • Flip (over an axis)
    • Multiply the x-term by a negative
Examples
  • Ex. Shifting Linear Functions
    • Notice how shifting UP by 5 is the same as shifting LEFT by 5.
      • Up (add 5 to the end of the equation):  F(x) = x becomes G(x) = x + 5
      • Left (subtract negative 5): F(x) = x becomes G(x) = (x – (-5)), simplified: G(x) = x + 5
    • Notice how shifting DOWN by 6 is the same as shifting RIGHT by 6
      • Down (subtract 5 from the end of the equation): F(x) = x becomes G(x) = x – 6
      • Right (subtract positive 6 from x): F(x) = x becomes G(x) = (x – 6), simplified: G(x) = x – 6

Alg2B 4.6 - Shift Up Down


  • Ex. Stretching and Compressing Linear Functions
    • Notice how stretching by 4 is the same as increasing the steepness of the slope
      • Stretching means multiplying the x by a number greater than 1
      • The coefficient of x > 1
    • Notice how compressing by \frac{1}{2} is the same as decreasing the steepness of the slope
      • Compressing means multiplying the x by a number between 0 and 1
      • The coefficient of x:  0 < x < 1

Alg2B 4.6 - Shift Stretch Compress


  • Ex. Multiple Transformations on One Graph

  • Ex. Shifting Quadratic Functions

Alg2B 4.6 - Shift Quadratic


  • Ex. Stretching and Compressing Quadratic Functions

Alg2B 4.6 - Stretch Compress Quadratic


  • Ex. Multiple Transformations on One Graph

  • Ex. Shifting Absolute Value Functions
    • Shifting Vertically: Up and Down: add or subtract the value from the end of the equation

Alg2B 4.6 - Shift Vertical Abs Value

 

    • Shifting Horizontally: Left and Right: subtract a positive (right) or negative (left) value from the x

Alg2B 4.6 - Shift Horz Abs Val


  • Ex. Stretching and Compressing Absolute Value Functions

Alg2B 4.6 - Stretch Compress Abs Val


  • Ex. Multiple Transformations on One Graph

 


  • Ex. Shifting Reciprocal Functions Vertically

Alg2B 4.6 - Shift Recip Fct


  • Ex. Shifting Reciprocal Functions Horizontally
    • Notice that the x-value (in the denominator) is affected

Alg2B 4.6 - Shift Recip Fct Horiz


  • Ex. Stretching and Compressing Reciprocal Functions
    • As expected, we multiply the x-term by the stretched or compressed value, then simplify

Alg2B 4.6 - Stretch Compress Recip Fx


  • Ex. Multiple Transformations on One Graph

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