Translations and Image Notation

The point $(3, 5)$ moves $4$ units right and $1$ unit down. Where does it land? Now do the same for $(-2, 0)$. What did you do to the coordinates both times?

A translation moves every point by the same horizontal amount and the same vertical amount, so the rule is an addition. The image of $P$ is written $P'$, read P dash, and it names the point $P$ has become.

When a transformation moves a point, the original and the new position both need names, and they need to be visibly related.

The convention is a dash. If the point $P$ moves, its image is called $P'$, read "P dash" or "P prime". Apply a second transformation and the next image is $P''$, read "P double dash".

For a shape the same convention applies to every vertex at once, so the image of triangle $ABC$ is triangle $A'B'C'$, with $A'$ being the image of $A$ specifically.

The pairing matters. Writing the image of $ABC$ as $B'A'C'$ would be a different labelling and would describe the vertices as having moved differently, so keep the order.

A translation slides the whole plane. Every point moves the same distance in the same direction, so the arrows joining points to their images are all parallel and all the same length.

Because nothing is stretched or turned, the image is congruent to the original: the same lengths, the same angles, and the same orientation. A triangle labelled clockwise stays labelled clockwise.

Unless the translation moves nothing at all, there are no invariant points. Every single point of the plane ends up somewhere else, which is not true of the reflections and rotations coming next.

A translation is described by two numbers, the horizontal and vertical amounts, and those two numbers determine it completely.

Moving $a$ units horizontally changes the x-coordinate by $a$. Moving $b$ units vertically changes the y-coordinate by $b$. The two are independent, so

$P(x, y) \to P'(x + a, y + b)$

Signs follow the usual convention: $a$ is positive to the right and negative to the left; $b$ is positive up and negative down.

So a translation of $5$ right and $2$ down has $a = 5$ and $b = -2$, and sends $A(-3, 4)$ to $A'(-3 + 5, 4 - 2) = A'(2, 2)$.

Write the signed values of $a$ and $b$ down before substituting. Almost every error in this topic is a sign taken from the words "down" or "left" and then lost.

Apply a translation of $(a_1, b_1)$ and then one of $(a_2, b_2)$. The first sends $(x, y)$ to $(x + a_1, y + b_1)$, and the second sends that to

$(x + a_1 + a_2,\ y + b_1 + b_2)$

So the combined effect is a single translation of $(a_1 + a_2, b_1 + b_2)$. Two slides make one slide, and you can add the components rather than tracking the intermediate position.

The order makes no difference, because addition is commutative: $a_1 + a_2$ is the same as $a_2 + a_1$. This is a genuine property of translations rather than a general fact about transformations, and it fails for other pairs, as the next two lessons show.

The intermediate position still deserves its own name when a question asks for it. The point is $P'$ after the first translation and $P''$ after the second.

The reverse question gives you a point and its image and asks which translation did it.

Subtract. If $P(2, -1)$ maps to $P'(-4, 3)$, then $a = -4 - 2 = -6$ and $b = 3 - (-1) = 4$. The translation is $6$ units left and $4$ units up.

One pair of points is enough, because a translation is determined by its two components and every point moves identically. Checking a second pair is nonetheless worthwhile when a whole shape is given: if the second pair gives different components, the transformation was not a translation at all.

That check is the practical test for whether a movement is a translation, and it is one subtraction per vertex.

Five items. Write the signed components before substituting in each case.

1. 1 Translate $(3, 5)$ by $4$ right and $1$ down.

   Show answer
   $a = 4$, $b = -1$.$(7, 4)$

2. 2 Translate $(-2, 0)$ by $3$ left and $6$ up.

   Show answer
   $a = -3$, $b = 6$.$(-5, 6)$

3. 3 $A(1, -4)$ maps to $A'(1, 2)$. Describe the translation.

   Show answer
   The x-coordinate is unchanged, so the move is purely vertical.$6$ units up

4. 4 A translation of $(2, -3)$ is followed by one of $(-5, 7)$. Give the single equivalent translation.

   Show answer
   Add the components separately.$(-3, 4)$

5. 5 How many invariant points does a translation of $(0, 4)$ have?

   Show answer
   Every point moves $4$ units up, so nothing stays put.None