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Horizontal and Vertical Translation

Horizontal and Vertical Translation

Author: Kitty Black

Y= a [ b ( x - h ) ] + k

Be able to use and understand 'k' and 'h'

The equation for transformation is :

Y= a [ b ( x -/+ h ) ] +/- k

Each variable has its own job in that equation. For starters we will learn about h and k.

h is used for horizontal transition either left or right.

k is used for vertical transition either up or down.


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  • A translation can move the graph of a function up or down ( Vertical Translation ) and right or left ( Horizontal Translation ).
  • A vertical translation of function y=f(x) by k units is written in either y-k=f(x) or y=f(x)+k.
    • ​If k>0, then the graph moves up.
      • ​y=f(x)+4, the graph ends up moving 4 units up.
    • If k<0, then the graph moves down.
      • ​y=f(x)-3, the graph ends up moving 3 units down.
  • A horizontal translation of function y=f(x) by h units is written y=f(x-h).
    • ​If h>0, then the graph moves right.
      • ​y=f(x-6), the graph ends up moving 6 units to the right.
    • If h<0, then the graph moves left.
      • ​y=f(x+3), the graph ends up moving 3 units to the left.
  • Vertical and Horizontal translations can be combines to form y=f(x-h)+k
    • ​y=f(x-3)+2
      • ​This causes the graph to move 3 units to the right, and 2 units up.