Translations and reflections
A translation is a slide
A translation moves every point the same distance in the same direction.
The shape does not turn or flip, so the image is congruent to the original and the same way up. The movement is given as a column vector: the top number is the movement across (right is positive), the bottom number is the movement up (up is positive).
The vector 4−2 means 4 right and 2 down.
- P is (−1, 3); translate it by 4 across and −2 up
- x: −1 + 4 = 3
- y: 3 + (−2) = 1
- P′ is (3, 1)
To find the vector of a translation, pick one vertex and its image and count from the original to the image. Counting from the image back gives the vector with both signs wrong.
A reflection is a flip
Every point goes to the same distance on the other side of the mirror line, measured at right angles to the line.
Points on the line stay where they are.
- x = a is a vertical line; y = b is a horizontal line. The y-axis is x = 0 and the x-axis is y = 0.
- Reflecting in y = x swaps the coordinates: (2, 5) goes to (5, 2).
- Reflecting in y = −x swaps them and changes both signs: (2, 5) goes to (−5, −2).
To describe a reflection fully, give the word reflection and the equation of the mirror line. To describe a translation fully, give the word translation and the column vector.
The line people mix up
The line x = 3 runs up the page through every point with x-coordinate 3.
Because the line x = 3 is vertical, reflecting in it changes x and leaves y alone. Many people draw x = 3 across the page instead. Check a point: (3, 0) and (3, 5) are both on x = 3, so the line is vertical.
Watch a point move
Open the graph transformations simulation and choose Solve for, then Image of a point. Type a = 1, b = 1, h = 4 and k = −3, and the point (2, 5). The image is (6, 2): a translation by 4 across and −3 up. Now set h = 0 and k = 0 and change a to −1. Where does (2, 5) go, and which line has it been reflected in? Then put a back to 1 and set b = −1 instead.
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