Lesson video

Hello, my name is Mrs. Buckmire.

Today, I'll be teaching you about plotting quadratics.

So make sure you have something to write with and something to write on.

It could be useful if you have a pencil and rubber because we're doing lots of graph drawing sometimes, you know you might make mistakes.

It might be easier if you can rub it out.

But no worries if not.

And remember pause the video whenever you need to and also pause and I asked you to, so I might want you to have a go at some things, so please do pause and have a little go at it because that's what's best for your learning.

And finally, remember you can rewind it.

If here's something that you don't understand or you don't quite catch what I'm saying, then just rewind it and that can be really useful at times.

Okay, let's begin.

So here, I want you to tell me what's the same, what's different? So both equation you've got a table of values and I've also got the graphs.

So pause the video and tell me what's the same and what's different.

Okay, so how did you do, what did you think about? So what were the same? Okay, yeah, they both have y equals, what else? They both had definite coordinates zero zero and we can see that from the table.

Oops, we can see that from the table and we can see that from the graph, can't we? So here we have it from the table and on the graph, it goes through, what's another word for zeros zero, it begin with O yes, the origin.

And it was also both go through one, one, so here and here and we can see it again on the table as well, anything else? That I've have an x in it? Yeah, true, okay yes they're both from blue, right? What is different then? Excellent, so this one is a linear and this one is a curve.

So this one's a straight line and this one is a curve.

So actually we call this one a quadratic.

Okay, do you say anything else? Yes, this one as x increases by one, y increased by same amount and this one that's steps aren't all equal.

Actually in this direction, as x increased by one y is decreasing this direction as x increases by one y is increasing.

So yes, very different in that way.

And that about the steps is as we move up each step, we increase by the same amount, that's what makes them a linear graph, linear.

So well done if you notice that.

Okay, so let's start connecting this together.

So let's find the missing coordinates here.

So we have y equals x squared, I want you to pause and fill in all the missing gaps.

Okay, so first let's start here.

So zero, when x equals zero what is y? Good zero, when x equals one what is y? Great what is in this, how do say this? Good yes x squared, so that's x squared, what does it mean? Yes, x times x , okay, so one times one is one.

So yes, negative one times negative one.

Good yes that's one as well.

Negative two squared, so I'd write it like this.

It is negative two times negative two, positive four.

Oh, here it gives me the y value, what's the x value.

What times itself equals to nine? Good three wait there's another one, what would that be? Excellent, negative three.

So remember whenever we square where we could actually, when we square it positive and we actually could get two values, we get a positive value and a negative value.

Nice, so these are different coordinates, is there anything else you notice? Yeah, so four x equals one and x equal to negative one, they both made a pretty one.

So this connect is hinting that the symmetry that quadratics have, let's look a bit further.

Okay, so we can write a quadratic in the form of y equals a x squared for us b x plus c.

I want to just break apart the line that symmetry, that exists here and kind of see it in this quadratic even, but actually that would be the line of symmetry there.

Now what makes it a quadratic is the highest power of x is two.

So it's x to a power of two is the highest power.

There's an x here, but that's the power of one.

If you imagine an x upon zero here which just equals one.

So actually x squared is the highest power.

And we're going to do a bit of exploring of this graph.

So what you can see this is, I'm using GeoGebra by the way.

So it's an awesome, awesome website our math master have created this for me to use.

So what I can do is I can actually vary a, so I can move a, I can make a bigger and let's see what happens when I make a bigger.

Okay, so that's when you make a bigger, what do you think is going to happen then if I make it smaller? Good, so when I made a bigger it became a bit thin, I didn't know and I'm going to make it small and it kind of widens out again.

What do you think is going to happen when it equals zero? Well, then the x squared are going to disappear, so what's going to happen? Great before I do it.

Excellent, so when it becomes zero, it becomes a straight line.

Because as we can see there below is y equals three x take away two.

And let's make a negative, what do you think it's going to happen there? X maybe becomes a quadratic again and actually what's happens going from positive to a negative? Yes, it's actually become an upside down.

Somebody would like to think there's a sad face.

So they think these are called positive quadratics when a is greater than zero and then when a less than zero, it becomes negative quadratics, they're kind of a sad face.

We can see what's happened in there.

Nice, okay lets good to be back on two was it or not? Two, okay, so what, what about b? I'm going to move b, I'm going to make it bigger.

What do think is going to happen? Oh, it's kind of been moving, isn't it? So as I move down, it's moving I want to make b smaller and it's moving up and as I make b bigger it's kind of moving down, it's interesting.

Let's see if I make it negative.

Oh it's moving kind of the other way.

What about when b is zero, did you spot it? When is b zero, it looks like that.

So when b zero actually the line of symmetry is here, the y axis, okay? So what about let's put, b back on was it at one? Lets put it on one, what about c, what happens? Having a prediction what's going to happen? I'm going to make c more negative first.

Have a little guess what could happen? Ah, it moved downward did you guess that? Okay, so as I get to zero, what do you think is going to happen? Let's see, zero, that's giving us anything particularly interesting.

Two x squared plus x, it goes through zero, zero That's for sure because now we don't have any extra.

So there's no, it's not a coefficient of x.

So wait, let me see here, so this is just adding this on.

So when it equals to zero, it means that now it's just, there's two x squared plus x or whatever.

There's a x squared plus b x there's no c, so definitely is always zero, zero because if x is zero then y must be zero also to say that.

So about when c becomes more positive, what do you think is going to happen based on what you've just seen? Excellent, I think it's going to move upwards.

Yes, so it looks like maybe there's a relationship between where it is on the y axis, because where is that negative five? Is between negative four, negative six.

And then where is that negative 3.

5 is between negative two and negative four.

And negative two is at negative two.

Oh, I think we spotted it, yeah, let's see if I move this.

Yeah, that's kind of where intercepts even the y axis that remains actually at 3.

5, doesn't it? What about if I move b, what about if I move a? Oh, look at that point, are you looking at this point here as the key point where you're looking at look is always going to be around 3.

5.

Remember how I changed a or how I change b, even when it's negative it still is 3.

5.

So if I want it to like go through at two and I changed c well, I'm going to change to two.

Excellent two, yeah, let me go.

But when you go through zero, it makes it easier.

If I want it to go through negative two, there you go, interesting.

Okay, so the things I want you to take from this is quadratics, we can think of them as happy face or sad faces, but really it's called a parabola, okay? So it's a curve, it's a type of curve.

And when you're sketching really want to make sure you're doing a curve, don't join with rulers, do that smooth curve.

And the other thing is that what we first say was about that line of symmetry, about that if I make, let's make a equal to one.

Let's just do something normal and you can see that line of symmetry is what we want to create, so let's go there.

There you go a x squared, yes.

So I had a line of symmetry along the y.

So you can see there's not always y c, if I change it.

So if I change it here to like x squared minus can it be minus four, does that work? Yes and now actually the line of symmetry is somewhere else.

It's not always y but it is easier way to some kind of line of symmetry going on.

Against a here's a line of symmetry, but that's one.

I got it, good, yes, okay? But let's get back to the other page.

Welcome back, it's time for your independent tasks.

What I want you to do is find a missing coordinates for each of these, I think I probably do this one first, then this one, then this one and this one.

But it doesn't really matter.

Be very careful with this one.

And I want you to list three things that you notice, anything you notice at all, okay? So pause the video and have a go now.

Okay, so how was that? So let's start with the one up here.

So it's y equals two x squared, so x equals one.

So it's two lots to x squared.

So two times one times one.

So you get to two and negative one squared is one.

So two times one, you also get to two.

But this one, the starred one the answer is eight.

So what times two, what squared and then times two equals to eight? Excellent, if you got two and that answer is zero.

Okay, the next one's coming up.

So we have two, two, three and one.

So that was x squared plus one and then the x squared plus x.

Six, four, zero and three quarters.

So a half squared or half times a half is one quarter and then you plus a half and you get three quarters.

So do pause and check those, make sure you've got that.

The most important thing really to remember is I think maybe some people might have made a mistake here on this one.

Let me go through this first.

So I think maybe here people might have made a mistake cause y equals x squared plus one.

So that means when you're subbing in, I would put that negative one in brackets and then square it plus one.

So negative one squared equals one, so actually that is one plus one.

So that's how you get that two.

So you've got that long, just make sure whenever you're square and maybe put the negative ones in brackets, you could even made mistake here.

So negative two squared is four plus negative two.

So four plus negative two is the same as four take away two, so that should equal to two.

So do just check those super carefully.

Okay and so you, how you can always see the answers there.

But for this one so negative x squared, so again I would actually use my brackets method.

So the fact that y equals minus brackets x squared, that is how you need to think about it.

So equals for one it's y equals minus of one squared, the negative of one squared, sorry.

So y equals negative one.

That's how you get negative one.

Here y equals negative one, the negative, sorry, of negative one squared.

So negative one squared is one so y equals the negative one, okay? So when it's still I think that's fine.

What's important is the negative of x squared.

Let me make it more clear here.

So y equals negative x squared, that is different to y equals two.

So this is not the same, not equal to negative x squared.

That's a completely different thing, okay? So our one is y equals to the negative x squared.

So what would negative nine, how do we get to that? Good, well done if you got three.

To this point take a minute to make sure all these were correct.

Okay, explore, so make a table of values for x, between negative four and four.

And then I want you to plot the graph on the same axes and the real explore and challenge is can you describe the transformation that maps one graph onto another, okay? So even though confident pause, I think all of you really just have a go at it and then I will give you some hints if you need it.

Okay, for my kids, so first you just have to set up the table, sometimes people find that hard.

When it says x is between negative four and four, I've gone up and steps off one here.

So x the smallest is negative four, the biggest is four.

We see that here and I've gone up in steps of one.

So you can complete these for each one and then so when I say transformation, what does that make you think of? Yeah, so I'm thinking maybe you could think of like, is there a translation between them, maybe a reflection, maybe a rotation, okay? So these all are symmetries as well that I'm thinking of.

So try and think actually how can get from once you have and hopefully if you put it on the same plot you'll kind of notice something or maybe even look on at the table, you might notice something, okay? Have a little go.

Okay, here are the answers.

So first I'd love you to check those carefully.

I'll give you a moment to do that.

So pause if you need to.

Okay, so already you might notice some things.

Yeah, what do you notice? That is interesting, so between x squared, y equal to x squared, so if it's negative where all the answer negative.

Maybe you don't even thought we didn't keep working out.

You were like, oh, it's just the same and just calm me down with the negatives.

Nice, oh do you notice anything else? If I compare this one and this one, what do you notice.

So comparing the y values.

Yeah, they're always one more so 16 plus one 17, nine plus one 10, four plus one five.

That's interesting because this one's y is equal x squared that's y equal to x squared plus one.

So already maybe you're thinking about what transformations.

Okay, let's do some sketching.

Once you're here plotted out really, I'm going to do a sketch.

A sketch is when you know it's around about it, it helps us to kind of explore things, but I'm not necessarily plotting each coordinate, but well done if you should have plotted it.

So I'm going to do this one in red and y equal to x squared I know is a positive quadratic.

So it's a happy face and it goes through, it touches at like zero.

So that was my best U-curve and it goes, it touches at zero zero.

Okay, I'm going to do this one in blue and to be fair, let's be good mathematicians.

Let's actually label our graphs like so everyone can see what it is.

So in maths to be a good mathematician you need to communicate your maths well, okay? So that's one of the things that helps other people understand what you're doing, okay? So y equals negative x squared actually should have looked a bit like this.

But I remember what I said you shouldn't be using a ruler.

You should actually be using like just a glass to kind of sketch around it and tha how it should look so let be a good mathematician and label this graph.

And finally x squared plus one, how did that look? I'm do it in green.

So firstly let me, I'm just going to find out when x equals zero, what was y? Y was one, okay? That's super helpful for me.

Cause that's telling me it's going to go through here on my y axis.

So I'm going to draw it like this and it's my best U-curve, oh, it's not the best, is it? like that but and that is y equals x squared plus one.

Okay, so our real question out, what we're interested in is how, what transformation maps one on to another? How did you do like do it? Like do it so you can see that to go from y equals x squared to y equals negative x squared, you could have a reflection.

So a flection in the x axis or in y equal zero.

Did you do a rotation? Oh, that would probably work a rotation about what? Yes, the origin, what's the degrees, what was the angle? Yes, 180 so you could actually also have a rotation, 180 degrees about the origin and then clockwise and anticlockwise wouldn't matter.

Okay, what about to get from y equal to x squared to y equals x squared plus one.

Nice, so to get from there to there, transform it.

Excellent, so really they've all actually been moved I know that I put the arrow to sideways, but actually if we were going to be correct, each coordinate has actually been moved upwards by one.

So you see this, it was originally at zero zero and now it's at zero one.

You know, your table we saw always it was plus one compared to it.

So actually it's moved up by one.

So that's the translation up one, do you remember how to do that notation? Yeah, so you can say some translation of excellent, zero, one that is one way.

And actually further one, we use that function notation to actually map out different transformations, which can be really useful.

Well done for having a go.

I really hope you enjoyed today's lesson and thank you so much for your hard work.

Do you please do the quiz, this is such a fantastic way for you to kind of assess what you understood from the lesson, there's feedback there.

So if there's anything that you get wrong, then I hope you've written some helpful hints and you can always come back to the video and look a little bit again.

Have a lovely day, bye.