Translating graphs

​​In a nutshell

The translation of a graph is a movement of the graph in either the horizontal or vertical direction. You can translate the graph of a function by modifying the function itself. Different modifications have different effects and it's important to know how to draw and interpret these translations.

Adding and subtracting constants

Adding and subtracting constants have different effects based on where in the function the constant has been included. Adding or subtracting a constant outside a function translates the graph vertically. Adding or subtracting a constant inside a function translates the graph horizontally.

The graph of $$y = f(x) +a$$​ is a translation of the graph $$y = f(x)$$ vertically by the constant $$a$$ in the positive direction, or by the vector $$\begin{pmatrix}0\\a\end{pmatrix}$$. 

Take the function $$f(x) = x^2$$:​

The graph of $$y = f(x+a)$$​ is a translation of the graph $$y=f(x)$$ horizontally by the constant $$a$$ in the negative direction, or by the vector $$\begin{pmatrix}-a\\0\end{pmatrix}$$.

Take the function $$f(x) = x^2$$:

Example

A function is given by $$f(x)=(x+6)^3$$. Draw the functions of $$y=f(x)$$ and $$y = f(x+2)$$ on the same axes and describe the transformation.

The graph of the function $$y = f(x+2)$$ is a translation of the graph $$y=f(x)$$ by the vector $$\begin{pmatrix}-2\\0\end{pmatrix}$$. If $$y=f(x)$$ is $$y=(x+6)^3$$, then $$y=f(x+2)$$ is $$y=(x+8)^3$$.

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