A translation adds to the coordinates. Moving 3 right and 2 up sends (x, y) to (x + 3, y + 2).
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Reflection changes one sign
Across the y-axis, (x, y) becomes (−x, y). Across the x-axis, it becomes (x, −y).
Rotation swaps them
A quarter turn counterclockwise about the origin sends (x, y) to (−y, x).
Check with one corner
Apply the rule to a single corner and see whether it lands where the picture says it should.
Step 2: Try It Yourself
The original stays on screen, so the movement is a relationship between two figures.
Every length and every angle is exactly as it was. The shape moved without changing, so the two figures are congruent.
Step 3: In Real Life
A video game
A game moves a character 5 right and 2 up by adding to its coordinates: (x + 5, y + 2). Flipping it to face left negates x. Every game engine runs on these rules.
Step 4: Watch an Example
One step at a time.
Watch Ana Predict a Reflection
A corner sits at (4, 3) and the shape is reflected across the y-axis.
- Step 1
A reflection across the y-axis changes the sign of x only.
Step 5: Your Turn
Practice makes it stick.
The Slide
Problem 1 of 2
(2, 5) is translated 4 right. What is the new x?
The Flip
Problem 2 of 2
(3, 7) is reflected across the x-axis. What is the new y?
Apply the Rule
1 of 4
(1, 2) translated 5 up. What is the new y?
2 of 4
(−3, 4) reflected across the y-axis. What is the new x?
3 of 4
(5, 0) reflected across the x-axis. What is the new y?
4 of 4
Does a translation change the size of a shape?
Step 6: Quick Check
Show what you know.
Question 1 of 1
(6, −2) translated 3 left. What is the new x?
What You Learned
- A translation adds the same amount to every point.
- A reflection flips the sign of one coordinate.
- A quarter turn about the origin swaps the coordinates and flips one sign.
For the grown-up
Translation adds to the coordinates — Sliding a figure 3 right and 2 up sends (x, y) to (x + 3, y + 2). Every point moves by the same amounts, which is exactly what makes the figure keep its shape. The rule is one addition per coordinate.
Reflection flips a sign — Reflecting across the x-axis sends (x, y) to (x, −y); across the y-axis it sends (x, y) to (−x, y). One coordinate keeps its value and the other becomes its opposite, which is precisely what crossing that axis means.
Rotation about the origin — A 90° counterclockwise turn about the origin sends (x, y) to (−y, x). A 180° turn sends it to (−x, −y). These rules can be discovered by plotting a point, turning the paper, and reading the new coordinates.
Why algebraic rules are worth having — Drawing works for one figure; a rule works for all of them and can be applied without a diagram. This is the move from geometry as drawing to geometry as computation, and it is what makes computer graphics possible.
Watch what happens to the numbers — Point (3, 5). Translate 2 right and 1 down. Add 2 to x, subtract 1 from y: (5, 4). Reflect (3, 5) across the x-axis: (3, −5). The y flips sign. Across the y-axis: (−3, 5). The x flips. Every rigid motion is a rule about the coordinates.