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 look at some of the keywords that we'll be using today.

First of all, let's compare different tansformations 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 was on the object itself and the second transformation on the initial image created from the first transformation.

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

Laura claims that the sense has changed, so there must have been some sort of reflection and that the size has changed, so there must also have been an enlargement.

However, without Jun giving any more information about those transformations, Laura cannot describe the specific details of what Jun did with any accuracy.

However, now Jun gives a little extra information.

He claims that the first transformation was a reflection in the line y equals nine.

Jun has now described the first transformation in full detail, so we can carry out that first transformation to figure out the details of the second transformation.

The line of reflection creates this middle image.

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

We can now conclude with the information on screen that the second transformation was an enlargement by a scale factor of two with a centre at one, eight.

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

Pause here to visualise or draw the first transformation, which is a 90-degree clockwise rotation around the centre of 1, 10 and then write down in full detail that second transformation.

Here's the middle image, and so we needed to translate that middle image by a vector of six, zero to get that final image.

Here Jun does not know where to start when identifying which transformations turned this object into that image and Laura is also confused, but for the opposite reason.

She has come up with lots of different possible transformations, but due to a lack of any extra information, she has no way of identifying which set of transformations are actually correct.

Sometimes there can be many different possible transformations from an object to an image and some descriptions may involve just one transformation whilst others may involve multiple transformations with the first transformation applied to the original object and subsequent transformations being applied to the most recently drawn image.

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

For example, the object could have been reflected and then translated, as we can see here, or reflected in a different line of reflection and then rotated, two different pairs of transformations that both result in that final image being in the exact same location.

Furthermore, Lauren notices that if you ever so slightly change 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 equals five, what if it was reflected in the line x equals four instead? Notice how that middle image changes location, meaning that the translation would have to be by the vector two, negative six rather than the original zero, negative six.

For this check, pause here to fully describe a possible set of two transformations, a reflection, then translation that could have turned that object into that image.

Okay, there are multiple correct answers.

Here's one of them, a reflection in the line y equals 10, and then a translation by the vector three, zero.

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

We could have had a translation by vector three, negative six, followed by a reflection in the line y equals seven.

Next check, the object was reflected and then translated to make this image.

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

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

For example, a translation by vector four, negative four or a reflection in the line y equals x.

For this same object image pair, pause here to write down a set of two transformations, however, one of those two must be a rotation.

Here's an example.

A 180 degree rotation with the centre of three, four, followed by a translation by vector one, negative one.

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.

Here are some of my suggestions.

Pause now to check through some of them here, Jun's alternate perspective on these problems is actually really sensible.

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

So rather than focusing on what is possible, sometimes it is more efficient to focus on which transformations are not possible.

And 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 of both object and image are the same.

However, Laura needs to be careful.

Her observation is only true if exactly two transformations have taken place.

If three or more transformations have occurred, then two rotations or two reflections may have been applied, which then resulted in that final image having the same orientation and sense as the original object, even if at some point in the middle of this set of transformations the orientation or sense was indeed different.

And knowing how many transformations has taken place really does help you to identify which transformations are and are not possible.

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

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

One must have been an enlargement and the other must have been a rotation since the orientation has changed by a quarter turn or 90 degrees.

The sense is the same between both object and image.

So exactly one reflection is not possible.

Here we have an object and an image.

Jun's observation seems pretty sensible.

There appears to have been a rotation.

However, here's Laura again, looking at it from a slightly unusual perspective that there's been an enlargement, not a rotation.

Jun dismisses Laura's idea since he notice that the object and image are the same size and this can only happen with a scale factor of one, right? And with a scale factor of one, the orientation should be the same whilst the object and image have different orientations.

However, Laura is absolutely correct.

Jun has forgotten about a different and very specific enlargement, an enlargement by the scale factor of negative one, which, for this example, has a centre of enlargement of three, five.

In fact, Laura's observation is an example of an equivalent transformation, no matter the shape of the original object.

An enlargement with a scale factor of negative one and centre of any pair of coordinates x, y is always equivalent to a rotation of precisely 180 degrees with a centre of rotation about that exact same pair of coordinates x, y.

The two transformations are not equivalent if the scale factor isn't negative one or the rotation isn't exactly 180 degrees or if the two centres are different from each other.

And for this check, the object has been transformed twice with the first being a translation.

Pause here to write down two different possible transformations for that second transformation.

The second transformation could have been a 180-degree rotation or an enlargement by a scale factor of negative one with either having the centre of six, eight.

Brilliant attention to detail so far.

Onto the practise task.

For question one, pause here to describe transformations between object A, image A prime and image B, and then plot and describe mappings to this new image, image C.

Next up, questions two and three.

Write down a single transformation from D to E and then analyse different equivalent transformations from object F to its image G.

Pause now for these two questions.

Great effort on identifying those equivalent transformations.

The answer to question 1a, well, examples include a translation by vector six, six or a reflection in the line x plus y equals 19.

For part B, the most straightforward single transformation is a reflection in the line x equals six, although there are other possible answers, including some rotations.

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

For question two, the single transformation is an enlargement by scale factor of two and centre zero, nine.

Finally, for question three, for part A, the equivalent transformation to an enlargement by a scale factor of negative one is a 180-degree rotation with the same centre.

And for this object, a second equivalent transformation is a translation by the vector negative 12, negative two.

And for 3b, two reflections are needed to retain the same sense and so a translation could have been the third transformation.

Pause here to check my example on screen now.

Next up, let's see whether we can identify invariant points after a set of composite transformations have been applied to an object.

Points on an object may stay in the same position after a transformation has occurred.

These are called invariant points.

Sometimes one point may be invariant, like with this reflection, whilst sometimes more than one point may be invariant like with this, with these two points after a reflection.

And sometimes a collection of points may be invariant, like with this reflection over an edge of that object.

However, invariant points do not stop there.

It is possible for an object to have no invariant points after one transformation has been applied, but for a point to return back to its original position after a second or third transformation is applied.

For example, how many invariant points are there after a translation to make this middle image, and then a rotation to get this final image? Well, there is exactly one invariant point at 3, 11.

Right, for this check, we have object B and a reflection and translation.

Pause here to identify the number of invariant points after these two transformations have been applied.

After the reflection and then the translation, we have two invariant points on two of the vertices of B.

Next up, pause here to identify the number of invariant points after a rotation, then Translation has been applied to C.

Let's focus on this vertex so we can gauge the orientation of C.

After a rotation, then a translation, there is exactly one invariant point at five, nine.

Furthermore, invariant points do not need to be either vertices or on the sides or edges of an object.

For example, how many invariant points are there after a reflection in this line for this middle image and then a translation to this final image? Notice that there is a collection of invariant points inside of the object along a segment of the line y equals nine.

That line is also part of the line of reflection from D, the object, to D double prime, the final image.

Right, for this check, pause here to identify the number of invariant points after a translation and then a reflection have been applied.

After a translation and then a reflection, there is a collection of invariant points, but where? Well, right here on a segment of the line y equals 10.

Sometimes a point may look like it is invariant after one or a set of transformations.

However, the point on the final image that overlaps the original object may actually be a different point.

For example, how many invariant points are there after a reflection in this line to get this middle image, then a rotation with this centre to get this final image? It looks like there is one invariant point, but we need to be very thorough to know for certain.

These two vertices on the original object and final image are actually not corresponding vertices.

Overlapping points must be corresponding to be invariant points, and so that single overlapping point is actually not an invariant point.

We can justify the claim that these two vertices are not corresponding by marking the vertex on the original object and each subsequent image, like so.

Therefore, after this reflection and this rotation, we can see that the vertex on the original object and its corresponding point on F double prime are not in the same location, so they definitely are not invariant points.

Using that information, pause here to identify the number of invariant points after a reflection, then translation have been applied.

After a reflection and then a translation, we can see that there is one vertex on the object and final image that overlap.

However, we can clearly see that these two vertices are not corresponding, so actually, there are no invariant points.

Amazing.

Let's put all of this information into action.

Pause here to give question one a go and identify the coordinates of any invariant points.

And for question two, object C is enlarged and then translated.

Izzy, Aisha and Lucas all claim different things about the number of invariant points in C after these two transformations, Pause now to show that they're all wrong.

Great effort on those two questions.

For the answers, pause here to check the locations of the images on screen for parts 1a and c.

And for 1b, we have one invariant point at six, seven, and for 1d, we have a collection of invariant points on a segment of the line x equals 11.

For 2a, we have the final image directly to the right of C with one shared edge.

And for 2b, there is actually only one invariant point at five, four in the midpoint of that line segment, even if it might look like there is a collection of them along the entirety of that shared edge.

Right, we've just looked at identifying invariants after some composite transformations, but what about creating transformations that result in some variants? Well, let's have a look.

So if you are given details of one transformation, it is possible to follow through and apply further transformations to the most recent image of an object so that the whole set of composite transformations at the end results in at least one invariant point with the original object.

For example, object A has been reflected.

What second transformation could I perform such that the final image has one single invariant point with object A? Well, there may be multiple possible transformations, each that result in a different invariant point.

Let's have a look at some examples.

A prime, our first image, could be translated by a vector zero, eight so that there is an invariant point at the coordinates four, eight.

We can identify that the two overlapping vertices are corresponding by I, so we can verify for certain that that is actually an invariant point at four, eight.

A different example could be that A prime is enlarged by a scale factor of negative one with a centre at 5.

54 to create this congruent image.

Whilst it may look like there is a collection of invariant points along that shared edge, this is not the case due to the sense of the image being different to object A.

However, there is one invariant point at the exact centre of that shared edge at 5.

5, 6.

This here, object B, is transformed by a rotation and then a translation.

The final image has one invariant point with object B.

Pause here to describe a possible translation vector that results in an invariant point and also write down the coordinates of that invariant point.

Here's the location of the image after that rotation, and here's an example of a cheeky solution.

The shared vertex at six, four was already an invariant point after the rotation, and so a translation by zero, zero keeps it at an invariant point, but if you're looking for a more conventional answer, a translation by zero, negative six gives an invariant point at three, four.

A final answer could be the vector negative three, six that gives an invariant point that's not on any vertex of B, this time at the coordinates 4.

5, 7.

Occasionally, a set of composite transformations results in the final image being in the exact same location as the original object.

This is called object invariance where the whole object is just made of invariant points.

The most common instance of this is when a transformation is followed by its inverse transformation.

For example, if we rotate object F by 90 degrees anti-clockwise and then rotate that image by 90 degrees, but clockwise this time with the same centre, well, we can see that the final image overlays the original object completely.

The inverse transformation to a rotation is just another rotation with the same centre and by the same angle, but rather than anticlockwise, it'll be clockwise or vice versa.

And for this check, which of these following sets of composite transformations result in the complete object invariance of object G? Is it A, a translation followed by the exact same translation? Is it B, a reflection followed by the exact same reflection? Is it C, a rotation followed by the same rotation but in the opposite direction? Or is it D, an enlargement by a scale factor of two, followed by an enlargement by a scale factor of 0.

5, but both with different centres? Pause now to identify any correct answers.

And the answers are B and C because they are a transformation followed immediately by the inverse transformation, which for reflection just so happens to be a reflection back again in the exact same line of reflection.

A isn't right because the inverse translation would have to be the vector negative one, negative three, not positive one, positive three, and whilst the second enlargement would result in the size of the final image being invariant with object G, it would be in a different location due to the two centres being different.

If the two centres were the same, then it would result in object invariance.

Amazing stuff, everyone.

Here are the final two practise questions.

For question one, we have this object that is reflected and then has further transformations applied.

Pause here to suggest possible transformations that satisfy these different invariance criteria.

And similarly for question two, pause here to suggest possible transformations for the object EFG.

Great effort on some complex transformations.

Here are the answers for question one.

part a, one example is a translation by vector negative two, zero.

For part B, we could have a rotation and then a translation or a pair of reflections.

Pause here to look at some of these in more detail.

For part C, for object invariance, we could apply the inverse transformation, which is just another reflection in the line of reflection x equals 10.

Onto question two.

Here is what the first image is after the negative enlargement.

For part A, we could have a translation by vector negative six, zero.

For B, we could have had a rotation.

And for C, we could have had a reflection.

Pause here to look at these answers in a little more detail.

And for part D, a pair of subsequent transformations could be either a pair of reflections or a rotation that reestablishes the original orientation and then a translation to return its location back to the original object.

And a massive well done, everyone on the effort that you have put into 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.

This includes a rotation of 180 degrees and an enlargement with scale factor of negative one being equivalent if they share the same centre.

We've seen that some objects have no invariance after one transformation, but may have one, multiple or a collection of invariant points after many transformations have been applied.

We've also seen object invariance where an entire shape is in the same location after a set of transformations.

And lastly, always check transformation descriptions by carrying them out.

I'm Mr. Gratton, and once again, thank you all so much for persevering with these fun transformation problems. Take care, everyone and have an amazing rest of your day.