Translation – MathMaine

Function Transformations: Translation

The red curve above is a “transformation” of the green one. It has been “translated” (or shifted) four units to the right. A translation is a change in position resulting from addition or subtraction, one that does not rotate or change the size or shape in any way.

Transformations are often easiest to analyze by focusing on how the location of specific points on the curve have changed. In the image above, the point $A_1$ on the green curve “corresponds” to point $A_2$ on the red curve. By this we mean that the transformation has moved point $A_1$ to $A_2$ .

Horizontal Translations

In looking at the coordinates of the two corresponding points identified in the graph above, you can see that the y-coordinate has not changed and the x-coordinate has increased by 4. If you were to examine any other pair of corresponding points, you would see the exact same difference. This can be described algebraically by the equation:

Using Corresponding Points to Determine Dilation Factors and Translation Amounts

Two earlier posts provide background information for this one: Function Translations and Function Dilations. If you are not already familiar with these topics, you may benefit from reading those first.

Given two points on a curve and their corresponding points after transformation, how does one determine the underlying transformations? Since two dilations and two translations may be taking place, it can be complex to try to separate the effects of dilation from those of translation.

As an example, consider the two curves above. The green curve is the graph of

$y_1(x_1)~=~(x_1 -1)^2+1$

and the red curve is a transformation of the green one. Two points are labeled on the green curve:

$A_1:(1,1)\\*~\\* B_1:(2,2)$

and their corresponding transformed points are labeled on the red curve:

Interactive Graphs for Linear, Quadratic, Rational, and Trig Functions Moved to GeoGebraTube

Some may have had trouble using my GeoGebra applets in their browsers. I have moved all of them to GeoGebraTube, which will hopefully fix the problem. You may search for them by typing “MathMaine” into the GeoGebraTube search box.

Links to all updated interactive graph applets are below. Comments and suggestions are always welcome!

Linear Functions

GeoGebraBook: Exploring Linear Functions,
which contains:

Interactive Linear Function Graph: Slope-Intercept Form

Interactive Linear Function Graph: Point-Slope Form

Function Translations: How to recognize and analyze them

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

$g(x)=x^2-6x+10\\*~\\*~~~~~~~=(x-3)^2+1$

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

$f(x)=x^2 ~~~~\text{translated right by 3, and up by 1}$

Describing $g(x)$ as a translation of a simpler-looking (and more familiar) function like $f(x)$ 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).