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4.4 – Shifting Functions

Objectives

  • Describe how the equation of a graph changes when it is shifted horizontally, vertically, or both.
  • Given a function and its graph, write the equation of a function for a graph that has been shifted from the original graph

 

Key Terms

  • Horizontally – To the left or right.
  • Shift – To move from one place to another.
  • Vertically – Up or down.

 

Notes

Shifting Functions
  • Shifting Functions involves adding positive or negative numbers to parts of parent functions.
    • This works for ALL parent functions: linear, quadratic, absolute value, reciprocal, cubic (x^3), quartic (x^4), etc.
    • Remember: adding a negative number is the SAME as subtracting a number.
Shifting Functions Vertically (Up or Down)
  • Up: to shift a graph up, increase the value of the y-coordinate for each point.
    • Video of a quadratic function shifted up
  • Down: to shift a graph down, decrease the value of the y-coordinate for each point.
    • Video of quadratic function shifted down

  • Examples: Shifting Graphs Vertically (Up or Down)
    • To shift a graph vertically, add positive numbers (UP) or negative numbers (DOWN) to the right-hand side of its equation.
      • Example: Quadratic
        • Parent: y=x^2
        • Shifted UP by 2: y=x^2+2
      • Example: Linear
        • Parent: y=x
        • Shifted DOWN by 4: y=x-4
      • Example: Quartic
        • Parent: f(x)=x^4+x^3
        • Shifted UP by 7: x^4+x^3+7
Shifting Functions Horizontally (Left or Right)
  • To shift a graph horizontally, use this formula y = (x – value)
    • Shift RIGHT: Subtracting a positive: (x – (+h)) becomes (x – h)
      • Positive h (in parenthesis above) stands for the horizontal shift to the right
    • Shift LEFT: Subtracting a negative (x – (-h)) becomes (x + h)
      • Negative h (in the parenthesis above) stands for a horizontal shift to the left
      • You will have to add parenthesis around the x-value and this new value.
  • Example: Quadratic Function Shifts (of the parent function): y=x^2
    • Shifted LEFT by 5: y=(x+5)^2
      • Notice how the plus 5 is INSIDE the parenthesis, before the exponent (square)
      • Shifting LEFT: (x + value).  It’s the opposite of what you would normally think!
    • Shifted RIGHT by 6: y=(x-6)^2
      • Notice that the minus 6 is INSIDE the parenthesis, before the exponent (square)
      • Shifting RIGHT: (x – value).  It’s the opposite of what you would normally think!

Alg2B 4.4 Shift Left Right

  • Review

Alg2B 4.4 Shifting Chart

Shifting BOTH: Vertically (Up or Down) AND Horizontally (Left or Right)
  • Original parent function: black
  • Shift Up and Right: red
  • Shift Down and left: blue

Alg2B 4.4 Shifting Up Down Left Right2


  • Example

Alg2B 4.4 Shifting Up Down Left Right

  • Notice how the red graph has shifted Down and to the Left
    • Identify a point (the vertex is an easy one to find), then:
      • Down by 2: add a negative to the end of the equation on the right side
      • Left by 1: add a value to the x-value, inside parenthesis
    • So, D is the answer: G(x)=(x+1)^2-2

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