# 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.
• 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!

• Review

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

• Example

• 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: $G(x)=(x+1)^2-2$