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All courses Math I · G-CO.2 37 of 59
Represent transformations as functions on points and compare distance/angle-preserving transformations with non-rigid transformations.

Write a coordinate rule for a translation

Problem
Write the coordinate rule that translates a point right 4 units and up 3 units.
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Hint

Handle the \(x\)- and \(y\)-coordinates separately: right changes \(x\), and up changes \(y\).

A translation adds or subtracts fixed amounts from the coordinates. Right means add to \(x\), left means subtract from \(x\), up means add to \(y\), and down means subtract from \(y\).

Solution walkthrough

01

Start with a general point

\[\text{input}~\text{point}~=~(x,~y)\]

A coordinate rule must describe the image of every point, so begin with arbitrary coordinates x and y.

02

Translate the horizontal direction

\[\text{right}~4:~x~\text{becomes}~x+4\]

The x-coordinate measures horizontal position. Moving right four units adds 4 to x.

03

Translate the vertical direction

\[\text{up}~3:~y~\text{becomes}~y+3\]

The y-coordinate measures vertical position. Moving up three units adds 3 to y.

04

Combine the coordinate changes

\[(x,~y)~→~(x~+~4,~y~+~3)\]

The first output coordinate records the four-unit right shift and the second records the three-unit upward shift.

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Another way

  1. Test the rule on (0,0): its image is (4,3), which is four units right and three units up.

!

Common mistake

Do not swap 4 and 3. Horizontal movement changes x, while vertical movement changes y.