TRANSLATIONS
Let’s start off with what you know….
We all know what a translation is right? In English, a translation is when you take a word or phrase from one language, and convert it into another one.
Makes sense?
But let’s dig a little deeper into what actually happens when you translate from one language to another.
A good example would be the word green.
In English, we know when we hear or read the word green, that means a specific color, but what about in other languages?
In Spanish, green is verde.
In French, green is vert or verte.
In Farsi, green is سبز which is pronounced as sabz.
All of these different words mean the same thing, so let’s look into where the word translate actually comes from.
Translate’s Etymology
So the word translate actually comes from the Latin word “translat” which means “to carry over”.
Makes sense, right?
When you translate a word or phrase from one language to another, you are actually “carrying over” that idea from one way of speaking, to another way of speaking.
This is what a translation is in math, you are “carrying over” a mathematical shape from one place, to another.
Now, I know you’re probably wondering, “This is a math class, why are we learning the history of some English word?”.
�Well, mainly because I want to build on your prior knowledge of the word so you don’t become confused. Unfortunately, the English language hadn’t developed when Geometry was beginning, so the ancient Greeks (you can blame Euclid) used Latin to explain different ideas and rules, and a translation is one of them.
So essentially, in math, a translation is just a movement
So to translate an object in math, you are taking it as it is, and moving it either up, down, left, right, forward, or backwards
without rotating, reflecting, or disfiguring the object.
So, a great example is this simple triangle:
As we can see, the triangles points are: (0,0), (0,4), (4,0)
Now let’s say we want to translate the triangle up by 3, how exactly would we do that?
Well, we need to find which part of the triangle controls the y axis.
That’s right! The y coordinates in each point indicate how high the triangle is. So how do you suppose we can move it up 3 spaces?
Add by 3!
So, our point of (0,0) becomes (0,3)
Our point of (0,4) becomes (0,7)
And finally our point of (4,0) becomes (4,3)
Now, let’s remember some of our Algebra 1 material
I mean, we know how to move up, right? We just add to the y values.
But how do we move to the left?
What about to the right?
What about moving down?
What if, we wanted to move our triangle (for whatever reason) down 7 units, and to the left 5?
Well, let’s look at our triangle first:
We know that the original points for this triangle are:
(0,0), (0,4), and (4,0)
We also know if we want to move it up 3 units, we need to add 3 to each y coordinate.
So how do we move down 7?
We subtract!
So, our new coordinates are: (0,-7), (0,-3), and (4,-7)
And here’s what it looks like!
We’re half way done
So, we’ve successfully moved the equation down 7 units, but what about moving it to the left 5 units?
We look at the coordinate that corresponds with going left and right, in this case, the x coordinate.
So, here’s what we left off with:
Now, if we want to move left on the graph, do we add or subtract?
We subtract!
So, our points were: (0,-7), (0,-3), and (4,-7)
But we need to move the triangle to left 5 units, so we need to subtract 5 from each x coordinate.
So, (0,-7) becomes (-5,-7)
(0,-3) becomes (-5,-3)
And (4,-7) becomes (-1, -7)
And now we have our triangle translated 7 units down and 5 units to the left.
That’s pretty much it!
So that’s how you translate objects on a graph.
Again, make sure you remember that translation just means moving it, not changing it at all.
When we want to change things, that’s a transformation, but we’ll get into that later.
Desmos activity
So, without further ado, here is your Desmos activity!