Translation of Axes
Simple Explanation
Translation of axes means shifting the origin of the coordinate system to a new location (h,k), without changing the direction of the axes. A point's new coordinates relate to its old ones by x' = x−h and y' = y−k.
Why Do We Need It?
Choosing a more convenient origin can dramatically simplify an equation — this is exactly how a translated parabola or circle equation can be converted back to its simpler standard form.
See It
A sketch showing the original x and y axes as dashed lines through the origin O, and a new set of axes drawn through a shifted origin O' at (3,-2), with a point P marked
Formula
Translation of Axes
x' = x−h, y' = y−k (new origin at (h,k) in the old system)
Converts a point's coordinates from an original (x,y) system to a new (x',y') system whose origin has been shifted to the point (h,k).
- (h,k)
- — the new origin's location, measured in the old coordinate system
- (x',y')
- — the point's coordinates in the new, shifted system
When to use it: Whenever it is more convenient to re-measure coordinates relative to a new, shifted origin.
Worked Example
Convert coordinates to a translated system
The origin is translated to the point (3,−2). Find the new coordinates (x',y') of the point (7,1) in the old system.
Why Does This Work?
Measuring a point's position relative to a new origin O' means subtracting that new origin's own position (h,k) from the point's original coordinates — exactly what x'=x−h and y'=y−k compute.
Real-Life Example
A surveyor switching to a more convenient reference point
A surveyor initially measures everything relative to a fixed starting marker, but finds it more convenient mid-project to re-measure relative to a landmark closer to the current work site.
Translation of axes gives every previously measured point's new coordinates relative to this more convenient reference point, without needing to re-survey anything.
Practice
If axes are translated so the new origin is at (2,5), find the new x'-coordinate of the point (9,5).
MediumCommon mistake
Adding h and k instead of subtracting, when converting FROM the old TO the new coordinates — x'=x−h, not x+h (addition is used only when converting back from new to old).
Quick Review
- x' = x−h, y' = y−k.
- (h,k) is the new origin's location in the old system.
- A more convenient origin can dramatically simplify an equation.