Translations of Graphs (Edexcel IGCSE Maths)

Spec point — 3.3B

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

Examples of translations

• A particular translation is specified by a translation vector

How do I translate graphs?

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

Vertical translations: y=f(x) + a

• $y=f(x)+a$ is a vertical translation by the vector $( \begin{gathered} 0 \\ a \end{gathered} )$

  • The graph moves up for positive values of $a$* *
  • The graph moves down for negative values of $a$
  • The x-coordinates stay the same

Example of a vertical translation

Horizontal translations: y=f(x + a)

• $y=f(x+a)$ is a horizontal translation by the vector $( \begin{gathered} - a \\ 0 \end{gathered} )$

  • 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

Example of a horizontal translation

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 $y = \mathrm{f} ( x - a )$ of the graph $y = \mathrm{f} ( x )$

  • $a$is subtracted from$x$ throughout the equation
  • Every instance of $x$ in the equation is replaced with $( x - a )$

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

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

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

    • $y = ( x - 6 )^{2} - 3 ( x - 6 ) + 7$ 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 $y = \mathrm{f} ( x ) + a$ of the graph $y = \mathrm{f} ( x )$

  • $a$ is added to the equation as a whole

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

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

  • 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

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

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

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

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

  • 2 is added to the equation as a whole

    • $y = 3 ( x + 1 )^{2} + 2$

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

  • the vertex is $( p , q )$
  • the value of $a$ does not affect the vertex coordinates

Worked Example The diagram below shows the graph of $y = \mathrm{f} ( x )$.

Sketch the graph of $y = \mathrm{f} ( x + 3 )$.

Answer:

The transformation of the graph is a horizontal translation with vector $( \begin{gathered} - 3 \\ 0 \end{gathered} )$ (3 units to the left)

The x-coordinates of the points change (subtract 3 from each) The y-coordinates of the points stay the same