Translations of Graphs

Translation Rule 1

For $y=f(x)+b$, the effect of $b$ is to translate the graph vertically through $b$ units.

  • If $b \gt 0$, it moves upwards.
  • If $b \lt 0$, it moves downwards.

Translation Rule 2

For $y=f(x-a)$, the effect of $a$ is to translate the graph horizontally through $a$ units.

  • If $a \gt 0$, it moves right.
  • If $a \lt 0$, it moves left.

Translation Rule 3

For $y=f(x-a)+b$, the graph is translated horizontally $a$ units and vertically $b$ units.

Example 1

Given $(x)=x^2$ is translated to $g(x)=(x-3)^2+2$, find the image of the point $(0,0)$ on $f(x)$.

$x-3$ means that the graph is to translate the graph $f(x)$ horizontally through $3$ units.
Thus the $x$-value $0$ transforms to $3$.
$(x-3)^2+2$ means that the graph is to translate the graph $f(x)$ vertically through $2$ units.
Thus the $y$-value $0$ transforms to $2$.
Therefore the point $(0,0)$ transforms $(3,2)$.

Example 2

Given $(x)=x^2$ is translated to $g(x)=(x-3)^2+2$, find the point of $f(x)$ which correspond to the point $(1,6)$ on $g(x)$.

$x-3$ means that the graph is to translate the graph $f(x)$ horizontally through $3$ units.
Thus the $x$-value $1$ is transformed from $-2$.
$(x-3)^2+2$ means that the graph is to translate the graph $f(x)$ vertically through $2$ units.
Thus the $y$-value $6$ transforms to $4$.
Therefore the point $(1,6)$ is transformed from $(-2,4)$.


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