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 .
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