Translating graphs

What is a translation?

A translation is basically when we move a graph (without changing its shape).

We can represent a translation using a vector of coordinates, where the top number represents the movement in the x direction, and the bottom number represents the movement in the y direction.

For example, the vector \begin{pmatrix}3 \\ -2\end{pmatrix} means we move 3 units to the right (positive x direction) and 2 units down (negative y direction).

Equation of a translated graph

If we have a function y = f(x), and we translate it using the vector \begin{pmatrix}a \\ b\end{pmatrix}, the equation of the translated graph will be: y = f(x - a) + b

This means we subtract a from x inside the function, and add b to the whole function.

The way to remember this is:

• Inside the bracket is x, and it’s the opposite of what the vector says
• Outside the bracket is the whole function, and it’s the same as what the vector says

There’s an easier way to think of this, though:

• If we want to move the graph in the x direction, we subtract from the x (which happens to be inside the function).
• If we want to move the graph in the y direction, we subtract from the y.

Using this, a translation of \begin{pmatrix}a \\ b\end{pmatrix} would give us y - b = f(x - a), which rearranges to the same equation as above: y = f(x - a) + b.

Example: find the equation of the graph y=4x + 1 translated by \begin{pmatrix}-2 \\ 3\end{pmatrix}

• Start with the originaal equation:
  • y = 4x + 1
• Translate using the vector \begin{pmatrix}-2 \\ 3\end{pmatrix}:
  • (y-3) = 4(x - (-2)) + 1
• Simplify:
  • y - 3 = 4(x + 2) + 1
  • y - 3 = 4x + 8 + 1
  • y - 3 = 4x + 9
  • y = 4x + 12
• Answer: y = 4x + 12.

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