Transformation of Axes by Translation: Formula and Examples

Transformation of axes by translation means shifting the coordinate system so that the origin moves to a new point (h, k) while the axes keep their old directions. If (x, y) are the old coordinates and (X, Y) the new ones, the relation is x = X + h and y = Y + k. The shape does not move — only the labels of points change.

This matters because many equations look messy only due to a badly placed origin. A circle written as x² + y² − 4x − 6y + 4 = 0 is really a circle centred at (2, 3); translating the origin there turns it into the clean X² + Y² = 9. Translation removes the linear terms, exposes symmetry, and is the first step in reducing second-degree equations to standard conic forms.

Key Concepts

• Translation formulas: x = X + h, y = Y + k (old = new + shift); equivalently X = x − h, Y = y − k.
• Invariants: distances, angles, slopes and the degree of an equation are unchanged — only the origin's location changes.
• Purpose: choose (h, k) to kill the first-degree terms in x and y of a second-degree equation.
• New origin: after translation, the point (h, k) in old coordinates becomes (0, 0) in new coordinates.

Formulas

• Coordinate relation: x = X + h, y = Y + k, where (h, k) is the old-coordinate position of the new origin.
• For a general second-degree curve ax² + 2hxy + by² + 2gx + 2fy + c = 0, the shift (h₀, k₀) that removes linear terms satisfies 2ah₀ + 2hk₀ + 2g = 0 and 2hh₀ + 2bk₀ + 2f = 0 (the centre of the conic).
• Units: h and k carry the same length units as x and y.
• Assumption: pure translation — no rotation and no scaling of the axes.

Worked Example

Translate the axes to the new origin (2, 2) and simplify 4x² + 9y² − 16x − 36y + 16 = 0.

1. Put x = X + 2 and y = Y + 2 into the equation.
2. Expand: 4(X + 2)² = 4X² + 16X + 16; 9(Y + 2)² = 9Y² + 36Y + 36; −16(X + 2) = −16X − 32; −36(Y + 2) = −36Y − 72.
3. Collect: (4X² + 16X + 16) + (9Y² + 36Y + 36) + (−16X − 32) + (−36Y − 72) + 16 = 4X² + 9Y² + (16X − 16X) + (36Y − 36Y) + (16 + 36 − 32 − 72 + 16).
4. The linear terms cancel and the constant becomes −36, so 4X² + 9Y² = 36, i.e. X²/9 + Y²/4 = 1.
5. Interpretation: the curve is an ellipse centred at the new origin (2, 2) in old coordinates, with semi-axes 3 and 2. Correct, since completing the square gives 4(x − 2)² + 9(y − 2)² = 36.

Common Misconceptions

• "Translation changes the shape of the graph." Only the coordinate labels change; the curve keeps its size, orientation and position relative to the world.
• "x = X − h is the translation formula." That is backwards: the OLD coordinate equals the NEW coordinate PLUS the shift, x = X + h.
• "Translation can also remove the xy term." No — removing the cross term needs rotation of axes, not translation.

Real-World Applications

• Conic reduction: every general second-degree equation is classified (circle, ellipse, parabola, hyperbola) by translating to its centre first.
• Physics: shifting to the centre-of-mass frame simplifies motion equations exactly like translating axes.
• CAD and graphics: objects are designed in local coordinates and translated into world coordinates for assembly.
• Statistics: centring data by subtracting the mean (a translation) is the first step of principal component analysis.

In short: translation re-homes the origin via x = X + h, y = Y + k, wiping out linear terms so the true symmetric form of a curve shows through. It changes descriptions, never geometry — and it is the standard opening move before identifying any conic.