Coordinate Geometry and Conics · Lesson 6

Translations, reflections and rotations on the grid

Translations slide every point the same way: (x,y)→(x+a,y+b)(x, y) \to (x + a, y + b). Moving right or up adds; moving left or down subtracts.

Reflections. Over the x-axis, (x,y)→(x,−y)(x, y) \to (x, -y). Over the y-axis, (x,y)→(−x,y)(x, y) \to (-x, y). Over the line y=xy = x, swap: (x,y)→(y,x)(x, y) \to (y, x). A reflection over an axis changes the sign of the other coordinate: reflecting over the x-axis keeps x and flips y.

Rotations about the origin. 90∘90^\circ counterclockwise: (x,y)→(−y,x)(x, y) \to (-y, x). 90∘90^\circ clockwise: (x,y)→(y,−x)(x, y) \to (y, -x). 180∘180^\circ: (x,y)→(−x,−y)(x, y) \to (-x, -y). To test a 90∘90^\circ rule, rotate (1,0)(1, 0): counterclockwise it must land on (0,1)(0, 1).

Two moves in a row. Do them in the order given and write down the point after each move. Order matters: translating and then rotating usually lands somewhere different from rotating and then translating.

Rules for moves on the coordinate grid
MoveImage of (x,y)(x, y)
Reflect over the x-axis(x,−y)(x, -y)
Reflect over the y-axis(−x,y)(-x, y)
Reflect over the line y=xy = x(y,x)(y, x)
Rotate 90∘90^\circ counterclockwise about the origin(−y,x)(-y, x)
Rotate 90∘90^\circ clockwise about the origin(y,−x)(y, -x)
Rotate 180∘180^\circ about the origin(−x,−y)(-x, -y)

Worked example Easy

In the standard (x, y) coordinate plane, point P is at (4,−3)(4, -3). Point P is reflected over the y-axis. What are the coordinates of the image of P?

  1. (−4,−3)(-4, -3)Answer
  2. (−4,3)(-4, 3)
  3. (−3,4)(-3, 4)
  4. (4,3)(4, 3)

How to solve it

  1. A reflection over the y-axis keeps the y-coordinate and changes the sign of the x-coordinate.
  2. (4,−3)→(−4,−3)(4, -3) \to (-4, -3).

Why each choice is right or wrong

  • A. Correct. The y-axis is the mirror, so the point moves straight across it: x changes from 4 to −4-4 and y stays −3-3.
  • B. Incorrect. (−4,3)(-4, 3) changes both signs, which is a 180∘180^\circ rotation about the origin, not a reflection.
  • C. Incorrect. (−3,4)(-3, 4) swaps the coordinates, which is the reflection over the line y=xy = x.
  • D. Incorrect. (4,3)(4, 3) reflects over the x-axis, which keeps x and flips y. Over the y-axis, x is the coordinate that changes.

Worked example Medium

In the standard (x, y) coordinate plane, point P at (2,5)(2, 5) is rotated 90∘90^\circ counterclockwise about the origin and then translated 6 units to the right. What are the coordinates of the final image?

  1. (−5,8)(-5, 8)
  2. (1,2)(1, 2)Answer
  3. (4,−5)(4, -5)
  4. (11,−2)(11, -2)

How to solve it

  1. Rotate 90∘90^\circ counterclockwise: (x,y)→(−y,x)(x, y) \to (-y, x), so (2,5)→(−5,2)(2, 5) \to (-5, 2).
  2. Translate 6 units right: (−5+6,2)=(1,2)(-5 + 6, 2) = (1, 2).

Why each choice is right or wrong

  • A. Incorrect. (−5,8)(-5, 8) translates first, to (8,5)(8, 5), and then rotates. Do the moves in the order given.
  • B. Correct. The rotation gives (−5,2)(-5, 2), and moving 6 units right gives (1,2)(1, 2).
  • C. Incorrect. (4,−5)(4, -5) rotates 180∘180^\circ, to (−2,−5)(-2, -5), before translating. A 90∘90^\circ turn swaps the coordinates.
  • D. Incorrect. (11,−2)(11, -2) uses the clockwise rule (y,−x)(y, -x), which gives (5,−2)(5, -2). Counterclockwise is (−y,x)(-y, x).

Trap Counterclockwise or clockwise, and the wrong axis

For 90∘90^\circ counterclockwise the rule is (−y,x)(-y, x); (y,−x)(y, -x) is clockwise. Both swap the coordinates, so both look right. Test with (1,0)(1, 0): counterclockwise it goes up to (0,1)(0, 1). For reflections, the x-axis keeps x and the y-axis keeps y.

Calculator When a calculator isn’t faster

Hand work is faster: each move is a one-line rule. If you test online and want a check, plot the original point and your answer in the built-in Desmos calculator and see whether the picture matches the move.