For the approach I now prefer to this topic, using transformation equations, please follow this link: Function Transformations: Translation

A function has been “translated” when it has been moved in a way that does not change its shape or rotate it in any way. A function can be translated either vertically, horizontally, or both. Other possible “transformations” of a function include dilation, reflection, and rotation.

Imagine a graph drawn on tracing paper or a transparency, then placed over a separate set of axes. If you move the graph left or right in the direction of the horizontal axis, without rotating it, you are “translating” the graph horizontally. Move the graph straight up or down in the direction of the vertical axis, and you are translating the graph vertically.

In the text that follows, we will explore how we know that the graph of a function like

which is the blue curve on the graph above, can be described as a translation of the graph of the green curve above:

Describing as a translation of a simpler-looking (and more familiar) function like makes it easier to understand and predict its behavior, and can make it easier to describe the behavior of complex-looking functions. Before you dive into the explanations below, you may wish to play around a bit with the green sliders for “h” and “k” in this Geogebra Applet to get a feel for what horizontal and vertical translations look like as they take place (the “a” slider dilates the function, as discussed in my Function Dilations post).

Continue reading Function Translations: How to recognize and analyze them