1.12
Graph of 2 parallel lines

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. Because

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.

A translated 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 by , the effect is a dilation in the direction by just this factor. This takes the line to .

Two lines: y=3x-1 and y=3/2(3x-1)=(9/2)x-3/2

If we multiply the coordinate of a point on the line by , the effect is a dilation in the direction by just this factor.

Two lines: y=3x-1 and y=6x-1

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 translate by in the direction with a dilation by in the direction? The line first goes to and then to . This works out to be .

The line y=(3/4)x+2

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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This article is from the free online course:

Maths for Humans: Linear, Quadratic & Inverse Relations

UNSW Sydney