
© UNSW Australia 2015
Transformations of lines
We now apply translations and dilations to lines, or more precisely to the equations of those lines. What happens to the equation of a line if we translate it by in the direction, or by in the direction? What happens if we apply a dilation goes to ?
In this step you will learn
-
how a translation in either the or direction affects the equation of a line
-
how a dilation affects the equation of a line.
Translating a line
If we take the line and translate it up by in the -direction, we get or just
If we take the same line and translate it by in the direction, then the equation changes more subtly to . Can you see why? Suppose the point lies on the original line. This means that
But then will lie on the translated line .
Q1 (E): Which of the following equations represents a translation of by in the negative direction?
a)
b)Q2 (M): Find the equation of the red line, and hence find the equation of its translate the blue line.
Summarizing, we have the following:
Replacing by in an equation represents a translation by in the direction.
Replacing by in an equation represents a translation by in the direction.
Scaling a line
If we multiply the coordinate of a point which lies on the line (in red) by , the effect is a dilation in the direction by just this factor. This takes the line to , which is the blue line below.
If we multiply the coordinate of a point on the line by , the effect is a dilation in the direction by just this factor.
Summarizing, we have the following:
Replacing by in an equation represents a dilation by in the direction.
Replacing by in an equation represents a dilation by in the direction.
Combining translation and dilation
So what happens if we combine a translation by in the direction with a dilation by in the direction? The line first goes to and then to . This works out to be .
Answers
A1. The translate by in the direction is b) . This simplifies to the equation .
A2. The point lies on the red line, and so its equation is of the form for some number . Since the point also lies on this line, this becomes . The blue line is the translate in the direction of the red line by , and therefore has equation or .
© UNSW Australia 2015