If you ask your language teacher to translate for you, you'll be quite happy with the results. But if you make the mistake of asking your math teacher to translate for you, you'll probably be rather concerned and confused when they pick you up and place you in another seat in the classroom.
What on earth could they be thinking? Well in Mathematics, translation is a type of transformation where you move something from one place to the other without resizing, rotating, or changing its shape in any way. So as long as your math teacher doesn't crush you or place you on your head, you did technically get what you asked for.
In this article, we will learn more about Translations.
Definition of Translation
Translation is the displacement of a figure from its original position to another, without a change in its size, shape or rotation.
With translation, a figure could be moved upward, downward, left, or right whilst the size is still the same. To have this done appropriately and accurately, it is done on a coordinate system.
The original object to be translated is called the pre-image, while the translated object is called the image.
Fig. 1. The pre-image is the original object, while the image is the new object after it has undergone translation.
What are the Rules for Translation?
Translating positively in the x-axis would shift the image to the right whilst translating negatively in the x-axis would shift the image to the left.
Translating positively in the y-axis would shift the image upward whilst translating negatively in the y-axis would shift the image downward.
The size and shape of the pre-image are the same as the image.
All the points on the coordinate system are shifted in the same amount of units and direction.
Given the triangle with coordinates \( A(-5, 7), B(0, 7), C(-5, 4) \).
Translate the given triangle 3 units up and 2 units to the right.
Solution
First, draw the original triangle by plotting the 3 corner points, and joining them with straight lines. This will give the triangle below.
Fig. 2. This is the pre-image of the triangle.
Next, move each of these points up by 3. The three points will become:
\[\begin{align} A(-5, 7) & \rightarrow A'(-5, 10), \\ B(0,7) & \rightarrow B'(0,10), \\ C(-5, 4) & \rightarrow C'(-5,7). \end{align} \]
This will give you the same triangle, just in a different place. Check that all the sides and proportions are the same length as before. Triangle A'B'C' below represents the vertically translated triangle.
Fig. 3. This is the pre-image (ABC) and the image of the vertically translated triangle (A'B'C').
Finally, shift each of the points in the triangle to the right by 2 points.
\[ \begin{align} A'(-5,10) & \rightarrow A''(-3,10), \\ B'(0,10) & \rightarrow B''(2,10), \\ C'(-5,7) & \rightarrow C''(-3,7). \end{align} \]
This will give the fully translated triangle, shown below.
Fig. 4. Triangle A''B''C'' above is the final triangle, the image of the translation. Triangle A'B'C is the triangle after vertical translation, and triangle ABC is the pre-image.
Translation Formula
Translating in Geometry is dependent on whether the figure is being translated vertically, horizontally, or both. Mathematically, the translation function is given by
\[ g(x) = f(x - k) + C, \]
where \(k\) denotes the number of units of translation on the \(x\)-axis, and \(C\) denotes the number of units of translation on the \(y\)-axis. This means that \(k\) is how much the figure has been shifted horizontally, and \(C\) is how much the figure has been shifted vertically.
Make sure to remember the negative sign before the \(k\) in the formula. Otherwise, you will end up translating your function in the wrong direction.
Types of Translation
In Geometry, shapes can be translated both horizontally and vertically. In either case, they can be moved in the positive or negative direction, and often you will have to combine both horizontal and vertical translations into one single transformation. Hence, it is important to understand how both work.
We recall the general formula for translations,
\[ g(x) = f(x - k) + C. \]
Horizontal Translation
For horizontal translation to occur, the above formula is replaced by: \[g(x)=f(x-k).\]
In this formula, \(k\) determines the amount that the function will be translated, and in which direction the transformation will be applied.
- If \(k>0\), then the translation is to the right (in the positive direction),
- If \(k<0\), then the translation is to the left (in the negative direction).
Vertical Translation
Vertical translation works in the same way as horizontal translation, but now we are looking at the value of \(C\) in the formula instead of the \(k\). The formula is:
\[ g(x) = f(x) + C. \]
The sign of \(C\) determines the direction of translation in the following way,
- If \(C>0\), then the translation is upwards.
- If \(C<0\), then the translation is downwards.
The general formula for translation mentioned earlier \[g(x)=f(x-k)+C,\] combines both horizontal and vertical translations.
The reasoning for the values of \(k\) and \(C\) is the same as the one mentioned in Horizontal translation and Vertical translation.
Examples of Translation
Let's first look at an example of translating a familiar function horizontally.
Sketch \(f(x) = x^2, g(x) = (x-2)^2\) and \( h(x) = (x+2)^2\).
Solution
First, sketch \(f(x) = x^2.\) You will probably have seen this function before, but if not, plug in some different values of \(x\) into the formula for \(f(x)\) to get a rough idea of what shape the graph will take. It should look like,
Fig. 5. The graph of \(f(x) = x^2\).
Next, you want to sketch \(g(x) = (x-2)^2 \). Remember the translation formula:
\[ g(x) = f(x - k) + C. \]
If you put \(x-2\) into \(f\), you will get \(f(x-2) = (x-2)^2 = g(x)\). Hence, \(k=2\) in our formula. Since it is \(k\) that is changed, it must be a horizontal translation. This means the graph of \(g(x)\) will be the same as the graph of \(f(x)\), but translated to the right by 2 points. The translation is done to the right, since \(k>0\).
Fig. 6. The graph of \(g(x) = (x-2)^2 \) is the same as the graph of \(f(x) = x^2\), but translated 2 units to the right.
Finally, you must sketch \(h(x) = (x+2)^2.\) Just as with \(g(x)\), this is a horizontal translation of size \(2\), but because there is a plus sign instead of a minus sign, it must be translated in the other direction, to the left. In fact, while identifying \(k\) in \(h(x)=f(x-(-2))^2\), we have \(k=-2\), and since \(k<0\), the translation is to the left.
Fig. 7. If you translate \(f(x) = x^2\) by 2 points to the right, you get \(g(x) = (x-2)^2\). Similarly, if you translate it 2 units to the left, you get \(h(x) = (x+2)^2 \).
Now, let's look at a similar example, but this time translating in the vertical direction.
Sketch \(f(x) = x^3, g(x) = x^3 + 1\) and \( h(x) = x^3 - 1.\)
Solution
First, sketch \( f(x) = x^3.\) If you have never seen this function before, start by writing down a table of corresponding \(x\) and \(y\) values to get a rough idea of what this should look like in your head. \(f(x)\) will look like,
Fig. 8. The sketch of \(f(x) = x^3.\)
Next, sketch \(g(x) = x^3 + 1\). Remember the translation formula,
\[ g(x) = f(x - k) + C. \]
Notice that this time, \(g(x) = x^3 + 1 = f(x) + 1.\) Hence, the \(C\) in our formula is equal to 1 in this case. Since it is the \(C\) that is changing, it is a vertical translation. Since \(C\) is positive, the translation must be upwards. This means that the function \(g(x) \) will be the same as \( f(x) \), but translated up by 1. This will look like,
Fig. 9. \(g(x) = x^3 + 1\) is the translation of \(f(x) = x^3\) upwards by 1.
Finally, you must sketch \(h(x).\) Again, this is the same as \(f(x)\) and has a translation of one, but the translation is downwards this time.
Fig. 10. \(h(x) = x^3 - 1\) is the translation of \(f(x) = x^3 \) down by one. Similarly, \(g(x) = x^3 + 1\) is the translation of \(f(x)\) upwards by 1.
Translation Rules
When working with points and coordinates, instead of using the Translation Formula, you should use the following translation rules.
When a point is moved right by \(k\), replace \(x\) with \(x + k\).
When a point is moved left by \(k\), replace \(x\) with \(x - k\).
When a point is moved upwards by \(C\), replace \(y\) with \( y + C\).
When a point is moved downwards by \(C\), replace \(y\) with \(y - C\).
To translate a whole shape, apply these rules to each of the points in the shape. Let's now look at some examples using these rules.
Examples using Translation Rules
First, let's look at an example of horizontal translation.
Translate the rectangle with the points \(A(-5, 7), B(2, 7), C(-5, 3), D(2,3) \) 5 units downward.
Solution
First, plot the rectangle. You can call this rectangle \(X.\)
Fig. 11. This is the original plot of the rectangle, the pre-image.
Remember that when a figure is translated by 5 units downward, the change is vertical and only in the \(y-\)coordinate. There is no change in the \(x-\)coordinates. By this, you must subtract 5 from all the \(y-\)component in each point.
Let \(X'\) be the translated image of \(X,\) then we have the application \( (x,y) \rightarrow (x, y-5) \). You can apply this to each of the points to get,
\[ \begin{align} A(-5, 7) & \rightarrow A'(-5,2) \\ B(2,7) & \rightarrow B'(2,2) \\ C(-5, 3) & \rightarrow C'(-5, -2) \\ D(2,3) & \rightarrow D'(2, -2). \end{align} \]
Hence, you have the coordinates of the translated rectangle. The graph of this new rectangle is below.
Fig. 12. The bottom rectangle (A'B'C'D') is the result of translating the top rectangle (ABCD) down by 5.
You will notice that the figure is translated, but the size of the image remains the same as the pre-image.
Let's look at an example that translates horizontally. This requires that only values on the \(x-\) axis are affected and not the values on the \(y-\)axis.
Translate the triangle with the points \( A(-6, 2), B(-4,4), C(-2,1) \) 4 units to the right.
Solution
Plot your points to form the triangle. Call this triangle \(X\).
Fig. 13. This is the original triangle, \(A(-6,2), B(-4,4), C(-2,1) \).
When a figure is to be translated to the right, the change is horizontal and only in the \(x-\) coordinate. There is no change in the \(y-\) coordinates. Hence, you must add 4 to the \(x-\) component in each point.
Let \(X'\) be translated image of \(X,\) then we have the application \( (x,y) \rightarrow (x+4, y) \). You can apply this to each of the points to get,
\[ \begin{align} A(-6,2) & \rightarrow A'(-2,2) \\ B(-4,4) & \rightarrow B'(0,4) \\ C(-2,1) & \rightarrow C'(2,1). \end{align} \]
These are the coordinates of the translated triangle. This is plotted below,
Fig. 14. Triangle A'B'C is the translation of triangle ABC by 4 points to the right.
However, as established that translation can be done horizontally, vertically or both. This is an example where a translation is applied in both directions at the same time.
A figure with the points \(A(-4,1), B(-4,3), C(-3,4), D(-1,2) \) is translated such that \( (x,y) \rightarrow (x+5, y+1) \). Sketch the original figure and the new figure after transformation.
Solution
First, sketch the initial figure.
Fig. 15. This is the polygon \( A(-4,1), B(-4,3), C(-3,4), D(-1,2).\)
Call this initial polygon \(X. \) Now, find the values for the image.
Let \(X'\) be translated image of \(X.\) Then \( (x,y) \rightarrow (x+5, y+ 1),\) as given. Hence,
\[ \begin{align} A(-4,1) & \rightarrow A'(1,2) \\ B(-4,3) & \rightarrow B'(1,4) \\ C(-3,4) & \rightarrow C'(2,5) \\ D(-1,2) & \rightarrow D'(4,3) \end{align} \]
These are the coordinates of the new figure. This figure is plotted below.
Fig. 16. Polygon A'B'C'D' is the translation of polygon ABCD up by 1 and right by 5.
It is also important to be able to work out what translation a figure has undergone, based on the image and pre-image.
In the below image, triangle ABC is the pre-image and triangle A'B'C' is the image. What has the triangle been translated by?
Fig. 17. The pre-image of the transformation is triangle ABC, and the image of the transformation is triangle A'B'C'.
Solution
We can draw a table and collect all the points for the pre-image (ABC) and the image (A'B'C').
Pre-image points | Image points |
A \((0,0)\) | A' \( (3,0) \) |
B \((3,-6)\) | B' \((6, -6)\) |
C \((5,-2)\) | C' \( (8,-2)\) |
You can see in the table above that in the image, every \(x\) component increases by 3, but the \(y\) values stay the same. This means that the triangle has been translated by 3 points to the right.
\[(x,y) \rightarrow (x+3,y). \]
Now, let's look at a similar problem, but with a translation in both the horizontal and vertical directions.
In the below image, triangle ABC is the pre-image and triangle A'B'C' is the image. What has the triangle been translated by?
Fig. 18. The pre-image of the transformation is triangle ABC, and the image of the transformation is triangle A'B'C'.
Solution
You can draw a table and collect all the points for the pre-image (ABC) and the image (A'B'C').
Pre-image points | Image points |
A \( (-8,4) \) | A' \( (-5, 3) \) |
B \((-5,5)\) | B' \((-2,4)\) |
C \((-4,1) \) | C' \( (-1,0)\) |
We can see from the table above that in the image, every \(x\) coordinate increases by 3, and the \(y\) coordinate is reduced by 1. This means that the triangle has been translated 3 points to the right and one point downwards.
\[ (x,y) \rightarrow (x+3, y-1). \]
Translations - Key takeaways
- Translation is the displacement of a figure from its original position to another, without a change in its size, shape or rotation.
- The pre-image of a translation is the original figure, and the image is the new figure once it has undergone translation.
- The translation formula is \( g(x) = f(x - k) + C, \) where \(C\) is the amount by which the shape has moved up, and \(k\) is the amount by which the image is moved right. If it is moved downwards or left, the values of \(k\) and \(C\) will be negative instead.
- The translation rules are:
When a point is moved right by \(k\), replace \(x\) with \(x + k\).
When a point is moved left by \(k\), replace \(x\) with \(x - k\).
When a point is moved upwards by \(C\), replace \(y\) with \( y + C\).
When a point is moved downwards by \(C\), replace \(y\) with \(y - C\).
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