Sketch Translations of Functions

Graphs can be translated horizontally and vertically.

To translate the graph horizontally, add or subtract a value before the function is applied. The movement is in the opposite direction to the value of the addition or subtraction. If $y = f(x)$ then $f(x + a)$ will cause the graph to be translated by $((-a),(0))$ .

To translate a graph vertically, add or subtract a value after the function is applied. That is, if $y=f(x)$ then $y=f(x)+a$ will result in a translation of $((0),(a))$ .

Example 1

Sketch the graph of $f(x)=x^2$ . Then sketch the graph of $f(x+4)$ .

Describe the translation.

To translate a function horizontally, apply the add/subtract before evaluating the function.

Answer: The line has been translated $((-4),(0))$

Graph showing translation amend value then function

Example 2

Sketch the graph of $f(x)=x^2$ . Then sketch the graph of $f(x)+4$ .

Describe the transformation.

To translate a function vertically, apply the add/subtract after evaluating the function.

Answer: The line has been translated $((0),(4))$

Graph showing translation function then amend answer