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A translation is a movement of the graph either horizontally parallel to the \(x\)-axis or vertically parallel to the \(y\)-axis.

The graph of \(f(x) = x^2\) is the same as the graph of \(y = x^2\). Writing graphs as functions in the form \(f(x)\) is useful when applying translations and reflections to graphs.

If \(f(x) = x^2\), then \(f(x) + a = x^2 + a\). Here we are adding \(a\) to the whole function.

The addition of the value \(a\) represents a vertical translation in the graph. If \(a\) is positive, the graph translates upwards. If \(a\) is negative, the graph translates downwards.

\(f(x) = x^2\)

\(f(x) + 3 = x^2 + 3\)

\(f(x) = x^2\)

\(f(x) - 2 = x^2 - 2\)

\(f(x) + a\) represents a translation through the vector \(\begin{pmatrix} 0 \\ a \end{pmatrix}\).

If \(f(x) = x^2\) then \(f(x + a) = (x + a)^2\).

Here we add \(a\) to \(x\), not to the whole function. This time we will get a horizontal translation. If \(a\) is positive then the graph will translate to the left. If the value of \(a\) is negative, then the graph will translate to the right.

\(f(x) = x^2\)

\(f(x + 3) = (x + 3)^2\)

\(f(x) = x^2\)

\(f(x - 2) = (x - 2)^2\)

\(f(x + a)\) represents a translation through the vector \(\begin{pmatrix} -a \\ 0 \end{pmatrix}\).