Translations of Graphs (DP IB Analysis & Approaches (AA): HL): Revision Note

Translations of graphs

What are translations of graphs?

• A translation is when:

  • the graph is moved (up or down, left or right) in the $xy$ plane

    • i.e. its position changes

  • but the shape, size, and orientation remain unchanged

• How far left/right or up/down is specified by a translation vector $\left(\begin{array}{c}x\\ y\end{array}\right)$:

  • $x$ is the horizontal displacement

    • Positive moves right

    • Negative moves left

  • $y$ is the vertical displacement

    • Positive moves up

    • Negative moves down

How do I find the graph equation after a horizontal translation?

• A horizontal translation of the graph $y=f\left(x\right)$ by $a$ units to the right, i.e. the vector $\left(\begin{array}{c}a\\ 0\end{array}\right)$, is represented by the equation

  • $y=f(x-a)$

• Any vertical asymptotes will also be translated

  • $x=k$ becomes $x=k+a$

  • Horizontal asymptotes stay the same

• To translate a graph $a$ units to the left

  • i.e. the vector $\left(\begin{array}{c}-a\\ 0\end{array}\right)$

    • the equation becomes $y=f\left(x+a\right)$

How do I find the graph equation after a vertical translation?

• A vertical translation of the graph$y=f\left(x\right)$ by $b$ units up, i.e. the vector $\left(\begin{array}{c}0\\ b\end{array}\right)$, is represented by

  • $y=f\left(x\right)+b$

• Similarly a vertical translation $b$ units down

  • i.e. the vector $\left(\begin{array}{c}0\\ -b\end{array}\right)$

    • has the equation$y=f\left(x\right)-b$

• Horizontal asymptotes change

  • $y=k$ becomes $y=k+b$

• Vertical asymptotes stay the same