Translation using vector notation

In the applet above, the graph of \(g(x)\) is the graph of \(f(x)\) translated 2 units along the x-axis and -1 unit along the y-axis. Each point on the graph of \(f(x)\) is translated by a vector \(\vec{v} \).

The graph of the function \(g(x)=f(x-h)+k\), is the graph of \(f(x)\) translated \(h\) units along the x-axis and \(k\) unit along the y-axis. Note the negative sign in front of \(h\).

Change the function to \(f(x)=\sin (x)\) in the input box in the applet above and observe the equation of \(g(x)\). Change the function to \(f(x)=1/x\).

Using the tool Translate Object by Vector

If you insert a function and a vector (using the Vector tool), you can use the Translate tool to make a translated graph.

Explanation

The graph of \(g(x)=x^2+3\), is the graph of \(f(x)=x^2\) translated 3 units along the y-axis.

The graph of \(g(x)=(x+3)^2\), is the graph of \(f(x)=x^2\) translated -3 units along the x-axis.

In order to understand the negative sign when translating along the x-axis, write the new function as \(f(x+3)=(x+3)^2\). We know that this function has its vertex when the expression within the brackets is 0, i.e. when \(x+3=0 \Leftrightarrow x=-3 \).