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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 (), quartic (), 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 ycoordinate for each point.
 Video of a quadratic function shifted up
 Down: to shift a graph down, decrease the value of the ycoordinate 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 righthand side of its equation.
 Example: Quadratic
 Parent:
 Shifted UP by 2:
 Example: Linear
 Parent:
 Shifted DOWN by 4:
 Example: Quartic
 Parent:
 Shifted UP by 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 xvalue and this new value.

 Example: Quadratic Function Shifts (of the parent function):
 Shifted LEFT by 5:
 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:
 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!


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
 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 xvalue, inside parenthesis
 So, D is the answer:

Important!
Practice (Apex Study 4.4)
 Practice: Pgs 9, 10, 14, 20, 21, 26, 27
 2 quizzes
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