﻿ Transformations of lines

# Transformations of lines

Changing scale and translating are important transformations that don't change the essential shape of curves or functions, but change the algebra. 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 $$\normalsize{2}$$ in the $$\normalsize{x}$$ direction, or by $$\normalsize{-3}$$ in the $$\normalsize{y}$$ direction? What happens if we apply a dilation $$\normalsize{[x,y]}$$ goes to $$\normalsize{[rx,ry]}$$?
In this step you will learn
• how a translation in either the $$\normalsize{x}$$ or $$\normalsize{y}$$ direction affects the equation of a line
• how a dilation affects the equation of a line.

## Translating a line

If we take the line $$\normalsize{y=3x-1}$$ and translate it up by $$\normalsize{2}$$ in the $$\normalsize{y}$$-direction, we get $$\normalsize{y=3x-1+2}$$ or just
$\Large{y=3x+1.}$
If we take the same line $$\normalsize{y=3x-1}$$ and translate it by $$\normalsize{2}$$ in the $$\normalsize{x}$$ direction, then the equation changes more subtly to $$\normalsize{y=3(x-2)-1}$$. Can you see why? Suppose the point $$\normalsize{[r,s]}$$ lies on the original line. Because
${\Large s = 3r – 1 = 3( (r+2)-2) – 1}.$
Then $$\normalsize{[r+2,s]}$$ will lie on the translated line $${\normalsize y = 3(x-2)-1 = 3x-7}$$.
Q1 (E): Which of the following equations represents a translation of $$\normalsize{y=3x-1}$$ by $$\normalsize{3}$$ in the negative $$\normalsize{x}$$ direction?
a) $$\normalsize{y=3(x-3)-1}$$
b) $$\normalsize{y=3(x+3)-1}$$
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 $$\normalsize{y}$$ by $$\normalsize{y-k}$$ in an equation represents a translation by $$\normalsize{k}$$ in the $$\normalsize{y}$$ direction.
Replacing $$\normalsize{x}$$ by $$\normalsize{x-h}$$ in an equation represents a translation by $$\normalsize{h}$$ in the $$\normalsize{x}$$ direction.

## Scaling a line

If we multiply the $$\normalsize{y}$$ coordinate of a point $$\normalsize{[x,y]}$$ which lies on the line $$\normalsize{y=3x-1}$$ by $$\normalsize{\frac{3}{2}}$$, the effect is a dilation in the $$\normalsize{y}$$ direction by just this factor. This takes the line $$\normalsize{y=3x-1}$$ to $$\normalsize{y=\frac{9}{2}x-\frac{3}{2}}$$. If we multiply the $$\normalsize{x}$$ coordinate of a point $$\normalsize{[x,y]}$$ on the line $$\normalsize{y=3x-1}$$ by $$\normalsize{\frac{1}{2}}$$, the effect is a dilation in the $$\normalsize{x}$$ direction by just this factor. Summarizing, we have the following:
Replacing $$\normalsize{y}$$ by $$\normalsize{ry}$$ in an equation represents a dilation by $$\normalsize{\frac{1}{r}}$$ in the $$\normalsize{y}$$ direction.
Replacing $$\normalsize{x}$$ by $$\normalsize{sx}$$ in an equation represents a dilation by $$\normalsize{\frac{1}{s}}$$ in the $$\normalsize{x}$$ direction.

## Combining translation and dilation

So what happens if we combine a translate by $$\normalsize{3}$$ in the $$\normalsize{y}$$ direction with a dilation by $$\normalsize{4}$$ in the $$\normalsize{x}$$ direction? The line $$\normalsize{y=3x-1}$$ first goes to $$\normalsize{(y-3)=3x-1}$$ and then to $$\normalsize{(y-3)=3(x/4)-1}$$. This works out to be $$\normalsize{y=(3/4)x+2}$$. A1. The translate by $$\normalsize{-3}$$ in the $$\normalsize{x}$$ direction is b) $$\normalsize{y=3(x+3)-1}$$. This simplifies to the equation $$\normalsize{y=3x+8}$$.
A2. The point $$[0,5]$$ lies on the red line, and so its equation is of the form $$y=kx+5$$ for some number $$k$$. Since the point $$[-3,-4]$$ also lies on this line, this becomes $$y=3x+5$$. The blue line is the translate in the $$x$$ direction of the red line by $$\normalsize{4}$$, and therefore has equation $$y=3(x-4)+5$$ or $$y=3x-7$$.

#### Maths for Humans: Linear, Quadratic & Inverse Relations  