Translations of Graphs (Edexcel GCSE Maths: Higher): Revision Note

Translations of graphs

What are translations of graphs?

• The equation of a graph can be changed in certain ways

  • This has an effect on the graph

    • How a graph changes is called a graph transformation

• A translation is a type of graph transformation that shifts (moves) a graph (up or down, left or right) in the xy plane

  • The shape, size, and orientation of the graph remain unchanged

• A particular translation is specified by a translation vector

How do I translate graphs?

• Let $y=\mathrm{f}\left(x\right)$ be the equation of the original graph

Vertical translations: y=f(x) + a

• $y=\mathrm{f}\left(x\right)+a$ is a vertical translation by the vector $\left(\begin{array}{c}0\\ a\end{array}\right)$

  • The graph moves up for positive values of $a$ 

  • The graph moves down for negative values of $a$

  • The x-coordinates stay the same

Horizontal translations: y=f(x + a)

•  $y=\mathrm{f}(x+a)$ is a horizontal translation by the vector $\left(\begin{array}{c}-a\\ 0\end{array}\right)$

  • The graph moves left for positive values of $a$

    • This is often the opposite direction to which people expect

  • The graph moves right for negative values of $a$

  • The y-coordinates stay the same

What happens to asymptotes when a graph is translated?

• Any asymptotes of f(x) are also translated

  • An asymptote parallel to the direction of translation will not be affected

How does a translation affect the equation of the graph?

• For a horizontal translation $y=\mathrm{f}\left(x-a\right)$ of the graph $y=\mathrm{f}\left(x\right)$

  • $\mathit{a}$ is subtracted from $\mathit{x}$ throughout the equation

  • Every instance of $x$ in the equation is replaced with $\left(x-a\right)$

• E.g. the graph $y=x^2-3x+7$ undergoes a translation of 6 units to the right

  • $y=\mathrm{f}\left(x\right)$ becomes $y=\mathrm{f}\left(x-6\right)$

  • $x$ is replaced throughout the equation by $\left(x-6\right)$

    • $y={\left(x-6\right)}^2-3\left(x-6\right)+7$ is the new equation

  • The equation can be left in this form or expanded and simplified

    • $y=x^2-12x+36-3x+18+7$

    • $y=x^2-15x+61$

• For a vertical translation $y=\mathrm{f}\left(x\right)+a$ of the graph $y=\mathrm{f}\left(x\right)$

  • $\mathit{a}$ is added to the equation as a whole

• E.g. the graph $y=4x^2+2x+1$ undergoes a translation of 5 units down

  • $y=\mathrm{f}\left(x\right)$ becomes $y=\mathrm{f}\left(x\right)-5$

  • 5 is subtracted from the equation as a whole

    • $y=4x^2+2x+1-5$

  • The equation can be left in this form or simplified

    • $y=4x^2+2x-4$

How do I apply a combined translation?

• For a horizontal translation of $p$ units and vertical translation of $q$units combined

  • $y=\mathrm{f}\left(x\right)$ becomes $y=\mathrm{f}\left(x-p\right)+q$

• E.g. the graph $y=3x^2$ undergoes a translation of 2 units up and 1 unit to the left

  • $y=\mathrm{f}\left(x\right)$ will become $y=\mathrm{f}\left(x+1\right)+2$

  • $x$ is replaced throughout the equation by $\left(x+1\right)$

  • 2 is added to the equation as a whole

    • $y=3{\left(x+1\right)}^2+2$

• Note that when the equation is in the form $y=a{\left(x-p\right)}^2+q$

  • the vertex is $\left(p,q\right)$

  • the value of $a$ does not affect the vertex coordinates