Lesson video

Welcome, everyone, and thank you all so much for joining me, Mr. Gratton, for this problem-solving transformations lesson.

Today we will use your knowledge of transformations to solve problems involving transformations.

Pause here to have a quick check of some of the keywords that we'll be using today.

First of all, let's compare different transformations of the same object that result in different images in the same location that overlap each other completely.

Jun has performed two different transformations to turn that object into that image.

The first transformation on the original object and the second transformation on the initial image created.

Jun asks whether it is possible for anyone to figure out what those two transformations were.

Laura claims that, "The sense has changed so there must have been a reflection, and the size has changed so there must also have been an enlargement." However, without Jun giving any more specific information about those transformations, Laura can't describe the specific details of the reflection or the enlargement accurately.

Jun, however, then gives a little extra.

He claims, "The first transformation was a reflection in the line Y = 9." Jun has now described the first transformation in full detail, so we can carry out that first transformation to figure out more of the details of the second transformation.

The line of reflection creates this middle image.

Laura correctly identified the second transformation as an enlargement, so let's join corresponding vertices with lines, like so, to find that centre of enlargement.

The second transformation was an enlargement by a scale factor of 2 with a centre of 1, 8.

Okay, let's have a look at another double transformation.

The first transformation is a 90-degree clockwise rotation around a centre of 1, 10.

Pause here to visualise or draw the first transformation, and then write down in full detail the second transformation.

Here is the middle image, and so we need to translate that middle image by a vector of 6, 0 to get the final image.

So, Jun's not so confident now.

Jun has absolutely no idea what transformations were applied to turn that object into that image, and Laura is also confused but for a different reason.

She has come up with lots and lots of different possible transformations and, because of a lack of information, she has no way of identifying which possible set of transformations is correct.

Sometimes it is possible to describe how an object is transformed into its image in multiple different ways.

Some descriptions may involve one transformation, whilst others may involve multiple different transformations, with the first one being applied to the object and subsequent transformations being applied to the most recent image.

It is best practise to check whether the description of each transformation or set of transformations is correct by carrying out those transformations.

For example, this object could have been reflected in the line X = 5 and then translated by the vector 0, -6, or it could have been reflected in the line Y = 5 and then rotated 180 degrees with a centre of 5, 2.

Two different pairs of transformations that both result in the image being in the exact same location.

Furthermore, Laura notices that if you change, ever so slightly, what the first transformation is, then the second transformation will change slightly too.

For example, rather than the object being reflected in the line X = 5, what if it was X = 4 instead? Notice how that middle image changes location, meaning that the translation would have to be by the vector 2, -6 and not 0, -6.

For this check, we have an object that has been reflected and then translated.

Pause here to fully describe one possible set of two transformations that could have turned the object into that image.

One possible answer is a reflection in the line Y = 10 and then a translation with vector 3, 0.

Furthermore, pause here again to fully describe a different set of two transformations where the first transformation was a translation rather than a reflection.

We could have had a translation with a vector 3, -6 followed by a reflection in the line Y = 7.

For this next check, the object was reflected and then translated to make its image.

However, the location of the image was also possible with a single transformation.

Pause here to write down a possible single transformation that turns the object into that image.

Here are two different examples: a translation by vector 4, -4 or a reflection in the line Y = X.

For this final check, pause here to write down a pair of transformations, but one of the transformations must be a rotation.

For example, we could have had a 180-degree rotation with a centre of 3, 4 and then a translation by vector 1, -1.

And next up, time to get creative.

Pause here to come up with and fully describe as interesting a pair of transformations as possible, that could have turned that object into that image.

I'd love to hear how creative your answer was.

Pause here to have a look at some of my suggestions.

Right, Jun's alternate perspective is pretty sensible.

There are potentially loads and loads of different transformations that map an object onto its image.

This is especially true when you can apply sets of two, three, four, or more transformations.

So rather than focusing on what is possible, is there any way to know which transformations are not possible? Laura's understanding of the properties of transformations is here to help.

For this particular example, a reflection and rotation have not happened, since the orientation and sense are the same.

However, Laura needs to be careful.

If many transformations have occurred, rather than just two, then two rotations or two reflections may have been applied, which then result in the image having the same orientation and sense as the object, even if it had a different orientation or sense partway through that set of transformations.

And knowing how many transformations has taken place helps you identify which transformations are and are not possible.

For example, exactly two transformations of different types have occurred.

Pause here to place a tick by each transformation that could have taken place, and a cross by each transformation that could not have taken place.

With exactly two transformations of different types, one must be an enlargement; the other must also be a rotation, since the orientation has changed by 1/4 turn or 90 degrees.

The sense is the same between object and image, which is why a reflection is not possible.

Brilliant attention to detail so far.

Onto the practise task.

For question 1a, two transformations are applied to object A to become image B, with the first being a rotation.

By carrying out this rotation, identify and describe the second transformation.

For part B, the object and image stay in the same position but a single transformation occurred to map A to B, rather than the two transformations discussed in part A.

Write down a possible single transformation.

And for C, A is transformed to make a different image, C, by carrying out the two transformations, then write down a single transformation that would've had the same effect of mapping image B onto this new image, C.

Pause now for question one.

Next up, question two.

Write down a single translation or a single rotation equivalent to the two reflections that were applied to D.

And for question three, write down a single transformation from F to this new image, G.

Pause now for these two questions.

Great effort on identifying those equivalent transformations.

The answer to question 1a, examples include a translation by vector 6, 6 or a reflection in the line X + Y = 19.

For part B, the most straightforward single transformation is a reflection in the line X = 6, though there are other possible answers including rotations.

For part C, for shapes B mapped onto C we could have had a reflection in the line Y = 12 or a 90-degree anti-clockwise rotation with a centre of 13, 12.

For question 2a, the translation has a vector -16, 0.

For 2b, the rotation is a 180-degree rotation with a centre of 10, 4.

And finally, question three, the single transformation from F to G is an enlargement by a scale factor of 2 and a centre of 0, 9.

Next up, now that we can perform a set of transformations to an object to make its image, can we apply a set of similar transformations to make the image overlay the original object completely? Let's have a look.

Here we have A that is translated by vector 5, -7 to create B, but we also have B being translated by vector -5, 7 to create C.

What do you notice about object A and the final image C? Well, object A and image C are in the exact same location.

This is the same as saying that B has been mapped completely back on top of A.

Pause here to think about or discuss what you notice about the vectors used in both of these translations.

Each translation has the same magnitudes or distances.

Both horizontal components are 5 and both vertical components are 7, but each are in the opposite direction, noted by one being positive and the other being negative.

We can formalise this observation.

If an object is translated to a new location by a vector a, b, then it can be translated back to its original location by the vector -a, -b, like so.

For this check, object D has been translated to image E.

Pause now to write down both the transformation that maps D onto E and the translation that maps E back onto D.

For D to E, the vector is +2, -6, whilst E to D has the vector -2, +6.

For any object A and image B, there is a transformation that maps image B back onto A.

For example, A is enlarged with a centre of 0, 0, and by the scale factor of 2 to make image B.

What transformation maps B back onto A? Well, since we're looking at a transformation that results in a change of size, we are definitely still looking for an enlargement.

However, B to A means the shape is getting smaller, so its scale factor will be 1/2 not 2.

Notice how 1/2 and 2 are reciprocals of each other.

The ray will still pass through each pair of corresponding vertices, so they will still meet at that same centre of 0, 0.

And so, the only difference between the enlargements from A to B and B to A is that the scale factor is now the reciprocal of the other scale factor.

For this check, object F has been enlarged to make image G.

Pause here to write down the transformation that maps G back onto F.

All parts of the enlargement stay the same with the one exception of the scale factor, which is 1/3 rather than 3, where 1/3 is the reciprocal of 3.

Similarly, we can see the object H has been enlarged onto image I.

Pause here to write down both the transformation from H to I, and then the transformation from I back onto H.

Both are enlargements with centres 6, 12, for H to I the scale factor is 4, whilst I to H the scale factor is the reciprocal at 1/4.

Now that we've seen how to perform the inverse of a single transformation, what about the inverse of a set of multiple transformations? Well, if an image is created from multiple transformations, then there is a strategy to map the final image back onto the original object.

For example, object J is rotated 90 degrees clockwise with centre 6, 7 to make this middle image, J prime.

It is then reflected in the line Y = 10 to make the final image K.

In order to map K back onto J we need to apply each inverse transformation one by one, starting with the final transformation that was originally applied.

Starting with the inverse reflection, which is just another reflection in the exact same line of reflection, Y = 10, to make this reflection the middle image K prime.

And then we apply the inverse rotation, which is a rotation by the same angle and at the same centre, but in the opposite direction, so anti-clockwise rather than the original clockwise, to give us the original location of the original object J.

Pause here to think about or discuss which details change and which details stay the same when inverting a rotation or a reflection.

For this check, L has been reflected and then rotated to make M.

Pause now to write down in the correct order the pair of transformations that maps M back onto L.

We start with the last transformation, the rotation.

It'll still be a 90-degree rotation and still with the centre 8, 7, but we will rotate it anti-clockwise, the opposite direction to clockwise, and then we will reflect it in the line X = 8, the exact same line as in the original transformation.

The inverse transformation to reflection is to reflect it back again in the exact same line of reflection.

Okay, onto the last check.

N has been translated but no details on the vector are given, that middle image is then rotated to make the final image P.

Pause here to write down in the correct order the pair of transformations that maps P back onto N.

The inverse rotation is identical to the original rotation, since 180 degrees clockwise is identical to 180 degrees anti-clockwise.

The translation vector is then -1, 5.

Great stuff.

Onto the final practise task.

For question one, pause now to look at rotations and reflections, and then the inverse transformations to these when applied to this, object A.

And similarly, for question two, pause now to look at enlargements and translations, and then the inverse transformations of these when applied to that same object A.

And finally, question three.

F has a set of composite transformations applied to it to make image G.

Pause here to plot the location of G, and then describe the set of transformations and then a single transformation that maps G back onto F.

And a very well done on all of your effort in all of these transformations.

Here are the answers.

To question one the locations of image B, image C, and the line of reflection between B and C have all been drawn on screen.

The answer to part B: image B is rotated by 90 degrees clockwise with a centre of 7, 17.

For part D: C is reflected in the exact same line of reflection X = 7 to make the original object A.

And for part F: C is reflected in the line X + Y = 24 to make B.

For question two: the location of image D is on screen.

For part B: D is enlarged with a scale factor of 1/3 and a centre of enlargement of 0, 22 to make the original object A.

Now, the location of image E is on screen.

For part D: E is translated by the vector -10, 6.

And for part E: E is enlarged with a scale factor of 3 and a centre of 15, 13 to make D.

And lastly, question three: the middle images of F and then the final image G are on screen.

For part B: the transformations are an enlargement, but this time with a scale factor of 1/2, and then a reflection in the exact same line Y = 13, and then a rotation that is 90 degrees anti-clockwise this time, and then again a final reflection that is the exact same one, the line Y = X + 1.

For part C: we have an enlargement by a scale factor of 1/2 with a centre at that shared vertex of 6, 1.

Thank you all so much for persevering with all of those tricky transformation problems in a lesson where we have identified that one transformation, or even a set of composite transformations, may be equivalent to a different transformation or set of transformations.

We've also looked at inverse transformations that map an image back onto its original object, where there are different inverse transformations for translations, rotations, reflections, and enlargements.

And lastly, we've seen that a set of composite transformations can be undone by applying the inverse transformations in the opposite order.

Once again, well done on your perseverance during this lesson.

Thank you all for joining me, Mr. Gratton.

Have an amazing rest of your day.

Take care, and goodbye.