Lesson video

Hello, my name is Mrs Buckmire and today I'll be teaching you about sketching quadratics.

So you need something to write with and something to write on.

If you can write with a pencil, that might be ideal so you can rub it out if you make a mistake, if you have a rubber.

But it's fine if you only have a pen, more than okay, I only have my computer pen.

Please do pause the video whenever I ask you to but also whenever you need to, go at your own pace and remember you can rewind the video as well so if you need to hear something again.

Can be useful, just rewind it.

Let's start.

Okay, so for your try this I want you to factorise the following quadratics, so you have three quadratics there, I know they're quadratics because the highest power of X is two.

So it's X squared and I want you to plot the quadratics for X between negative five and five.

So first factorise and then plot it.

When you plot, you might want to do a table, so where X goes from negative five to five, substitute them in and then really what I'm most interested in, what do you notice? So from the three graphs from the factorizations, what do you notice? Pause the video and have a go.

Okay, so when we factorise what did we get? X plus two X plus one, yes? Next one, ooh looks similar but now we have a negative three X so it's going to be X take away two, X plus, X take away one even.

So X take away two X take away one.

And final one, X plus two, X take away one.

Okay, for the graphs I've used Desmos.

So Desmos is a free graphing calculator, you can google that, super useful, but that's where I've got these images from, so here, here, and here.

So do pause the video and check the graphs look similar.

I've put in some key coordinates.

Okay, so you can see that I have put in this coordinate and this coordinate and this coordinate.

What do you notice about all the coordinates that I've put in? Yeah, they're all ones where either the X value, X ordinates even, or Y ordinates is zero.

So these values, I'm going to actually circle them.

These are called when the X is zero, they actually go through the Y axis, so you can see this is the Y axis that it's cutting through.

So these are the coordinate where the graph cuts through the Y axis and they're called the Y intercept.

That is a very bad straight line, but you know what I mean.

So they're called the Y intercept.

When X equals zero.

And then actually, when it goes through X axis, what do you think it's called? Good, it's actually called the X intercept.

And actually, they're also known as, like solutions of our Y equals zero.

So what do you notice there, tell me anything.

Yes, they're all our positive quadratics, so they all have just one in front of X squared.

So the coefficient of X squared is one, 'cause there's nothing there.

So that's how we're efficient in maths.

And so they all actually have this like smiley face, so this positive quadratics, similar shape for all of them.

Excellent, so the first one and second one both go through zero, two.

So both cross the Y axis at two, but then the last one crosses it at negative two, interesting.

Anything else? Yeah, so maybe thinking about these X intercepts.

So here we have negative two, zero, and negative one, zero.

Just looking at that X part.

So negative two and negative one and this is X plus two, X plus one.

One, two.

This was negative two, negative one.

Negative two, one.

This was plus two, negative one.

So you might notice like, it's as if, some people say change of signs.

It's like negative two, plus two become negative two.

Plus one become negative one.

Negative two became positive two.

Negative one became one.

Plus two became negative two Negative one became one.

So this actually is the zero product property.

So this isn't massive, this doesn't really matter but I'm going to show you this quickly because some of you might be interested.

So, like I said about solutions, if we want when Y equals zero, so therefore we have X plus two and X plus one, by the zero product rule, now what's product? Good, when we multiply.

So zero product rule.

Two numbers to make zero so when can you, what two numbers multiply to give zero? Yeah, there's loads.

But one of them must always be zero.

So here, either X plus two equals zero or X plus one equals zero.

And if you solve each one, you'd get negative two for the first one and negative one for the second one.

How awesome is that? So you can also do that for the other two graphs but that is not mainly what we're focusing on but I just thought, interesting thing to observe.

Okay, let's connect actually what we do want to learn and how we do want to sketch the graph.

So, what I want you to be thinking of, what I'm thinking of when I'm sketching a quadratic.

First thing, is it a positive or negative quadratic? So by that I mean is the coefficient of X squared positive or negative? Then I'm thinking, what is the Y intercept? So where does it cross my Y axis and I do that by subbing in X equals zero.

So, let's do a little axis.

I'm going to draw straight here and here, so there's my axis, my X axis, my Y axis.

So first, is it positive or negative? Well, it's positive.

What's the Y intercept? Y subbing in X equals zero.

So this is when X equals zero we have zero plus zero take away eight.

So Y equals negative eight.

So it's going to go through zero, negative eight, which is down here and that's the first thing I plot.

Now, I already know the shape because I know it's positive, so I know it's going to be kind of that shape.

So therefore I can then factorise it.

So I would then factorise it and think, okay two numbers that add to give two but multiply to give negative eight.

So negative eight, it could be four and negative two.

So, factorise equals to X plus four times X take away two.

Knowing that then, therefore I know that it's going to go through negative four and positive two.

Now these bits don't matter, so I'm not expecting you to do this, but I'd just like to show you the shape really well.

So there you go.

And this goes through a zero, negative eight.

So I like to put the zero in and write the whole coordinate just because it's more accurate than just writing negative eight.

But you can just write negative eight.

But that coordinate is zero, negative eight.

And that's the sketch, that's it.

It'd be even better if we actually label it to say Y equals to X squared plus two X take away eight.

But that's it.

So, let's do this together.

First, is it positive or negative? Good, it's a positive quadratic, so I'm just going to do my little smiley face here.

What's the Y intercept? I get zero take away zero plus 15.

So Y equals 15.

When I draw my graph I know it's going to go through here at zero, 15.

You can factorise it, and get X take away three, X take away five.

Just maybe pause it, check that I'm right? So by zero product rule, I know it's three and five, but don't worry if you haven't done that.

So therefore, the graph goes through, whoops , I'm so funny.

As if I have actually written five there.

Obviously, five is a positive number, and it's going to be over here.

Do you know why I made a mistake? Because it had to be a smiley face and how does that work out where a point is- if the point was here, then that smiley face wouldn't work, would it? That would be a sad face.

I was like, that's definitely wrong, and then I noticed three's bigger than five, Mrs. Buckmire, what are you doing? So, back to it.

This is our lovely quadratic.

I don't even like that.

There we go.

So that's what it looks like.

Okay? So the main thing that I want you to know is, it's positive, it's a smiley face.

Y intercept: find it.

Zero, 15.

Added bonus if you get these two.

Added bonus, but that's not what it's all about, okay? Right, let's do a quick check? So, which sketch could represent X squared take away X take away six? So you're thinking about positive and negative quadratics.

Thinking about the shape, thinking about the Y intercept.

And that's all I've done here.

It's to focus you on it.

Pause the video and have a go.

Okay, so was it positive or negative? Fantastic, it was positive.

And where did it go through- what was the Y intercept? Excellent, so zero take away zero take away six, so Y equals negative six.

So it went through a zero, negative six, and it's got a smiley face, so it must be D.

Well done.

Did some of you factorise it? Could you factorise it? Yes you could.

You could have X plus two and X take away three.

Therefore we know actually on this graph, this value- what did you get, if you did it? You extra people who went even further? Negative two, well done! And this one would be positive three.

So if you got those added bits, pushing ahead, well done.

Okay, Zaki wants to sketch this graph.

Is this a positive or negative quadratic? Find the value of Y when X equals zero.

Use your answer to A to plot where the graph crosses the Y axis.

Sketch the graph.

Now, something a little bit trickier, how would the graph be different if instead he had to sketch negative X squared take away five X take away six? Have a go at sketching that graph, okay? So this is page one of two.

It's the first independent task, and then there's another question as well.

Okay, the second question.

I want you to sketch the following graphs on the same axes.

What do you notice? So, Y equals X squared take away one.

Y equals negative X squared plus two.

Y equals X plus two times X take away two.

Pause the video and have a go at sketching these.

I don't want you plotting them, sketch them, thinking about your answers.

Okay, so I'm going to use Desmos calculator to quickly show you this sketch.

I am also going to show you E.

What you should have got- you should've got the Y intercept being zero, six.

And the green graph here at the top- It was a positive quadratic, it looks like this, it goes throughout zero, six, is what you should've got.

If you went fair and found a way across it, you might be able to see on this graph, it's at negative two, zero, and negative three, zero.

So pretty impressive, well done.

How a graph would be different? Well, what you might've noticed is this equals the negative of X squared plus five X plus six.

So when you did it you would've seen, oh, it's a negative quadratic, so it's a sad face.

And it goes through at negative six.

And actually what you might've seen, is it is, what's the word? What's the transformation from the first graph to the second graph? Yes, is a reflection in the X axis, so well done if you got that.

Even better if you labelled it, which I haven't done.

So that one was Y equals X squared plus five X plus six.

And this one is Y equals negative X squared, negative five X take away six.

So, negative X squared take away five X take away six.

For this first one, we have X squared take away one, so it's a positive quadratic.

So it looks positive, it's going to be a smiley face.

And Y intercept will be zero, negative one.

Is that what you got? Yeah, it goes through here, doesn't it? So this graph has been drawn using Desmos again.

Desmos graphing calculator.

Graphic calculator, even, super super helpful.

This is Y equals X squared take away one.

This next one gives us a negative quadratic, 'cause it has a negative in front of the X squared.

It goes through zero, two.

So actually it's our blue one here.

So I'm going to label it here.

And so this value here, this coordinate, is my zero, two.

Make sure you've labelled it, so tell yourself off if you haven't labelled it.

I mean I haven't here.

But I should.

I'm going to do it now.

Make sure you've labelled it.

'Cause that's how you show off- yeah I know this.

This one, it might be useful to expand it, and you get: X squared take away four? Maybe some of you guys are like, oh I recognise that.

Maybe it's got a special name.

What is it? Difference of two squares, and as does this one as well.

So, here it's a positive quadratic.

It goes throughout zero, negative four.

And we can see that here.

Zero, negative four.

So Y equals X squared take away four.

And what do you notice? Yeah, interesting.

We've only got one negative, but it looks like that negative has some kind of symmetries, maybe between the green one, 'cause they pass through this point.

So I wonder if maybe there's a line of symmetry, perhaps here.

Let's do it in purple? Maybe some kind of line of symmetry here.

It kind of looks like some of them have been shifted in different ways.

So maybe it be, like, this blue one, Y equals minus X squared plus two, look like the Y equals X squared take away one, maybe flipped over and moved up a bit.

And there's actually lots of different relationships between these graphs.

So one if you spotted any of that.

If you found any of these coordinates as well, hopefully you can see them on the screen, but the first one went through on the X axis one, zero and negative one, zero.

The Y one, the blue one, actually went through root two zero, and negative root two zero.

I doubt you would've got that, but well done if you can figure that out.

And the final one went through two, zero, and negative two, zero.

So well done if you did that extra bit of work as well.

Okay, so I want you to decide if the following statements are always, sometimes or never true.

Use specific example to justify each of your responses.

So if it's always true, try and kind of prove it.

If it's sometimes true, maybe show when it is and when it isn't.

And if it's never true, try and justify why, okay? Pause the video and have a go at this.

You might even be able to use some examples from today's lesson to help you out there.

Pause in three, two, one.

Okay, quadratic graphs cross the x-axis in two places.

That always, sometimes or never true? Good, it is actually sometimes true.

Here is an example of when it does not work.

So here, why does it not work? Good, it actually doesn't cross the x-axis at all.

And it's because if you're trying to solve this, so the x-axis is when Y equals zero, and that means that you'd have negative one equals X squared.

And that's imaginary numbers, and this is a real coordinate grid.

So what I mean by that is basically, you learn about it at A-level, but it doesn't exist at this current- in this grid, currently.

So, yes, this is only sometimes true.

Let's write sometimes.

So we have seen lots when there is, but here in the example when it isn't.

What about quadratic graphs cross the y-axis? That always, sometimes or never? I'm just going to draw a sketch.

Sometimes you might think, oh, yeah that could be never.

'Cause maybe you've got a really steep quadratic here like this.

But actually as we get further and further, eventually it will.

Eventually it will cross that y-axis.

Because it's always in the form of Y equals AXE squared plus BX plus C, that if it crosses the y-axis when X equals zero, then Y is going to always equal C.

Even if C is zero, that means that actually is just- When X equals zero, Y equals AXE squared plus BX plus C where A, B and C are some kind of variable, some kind of numbers, Y will equal C.

And even if it's zero, then it crosses the x-axis, there.

Then it crosses at the origin- it is at the origin, so it means it crosses the X and Y axes, here.

Or if it's not, there's always going to cross at zero, C.

So actually, yes, it's always, always, always, this was always true.

Okay, what about this one? Every quadratic has a minimum point.

Is that always, sometimes or never? Sometimes true.

So, when we have our smiley face, yeah, it has a minimum.

That's the lowest it's ever going to be.

But when we have a negative quadratic and our sad face that was positive, and here is negative.

Well, then actually, it will always get lower and lower.

So actually, there is no minium point.

It goes all the way down to negative infinity.

Yes, it does not always have a minimum point, so it is only sometimes true.

And finally: for all quadratics, as the X value increase the Y value also increases.

Interesting.

Do you maybe draw a graph, maybe make some tables.

This is another counter example.

So here, on this side, as X increases, so negative five, negative four, negative three, we can see that the Y value is increasing.

It's getting less and less negative.

It's going towards our zero.

But then here, on this side, as X increase in this way, Y is decreasing.

Y is getting lower and lower.

So as the X value increases, the Y does not increase, it decreases.

So this is only sometimes true.

And even then, here, as the X value increases on this side, the Y value is decreasing, and on the other side, yes it is true.

So it's only true on a graph at a particular point on a particular side.

So for positive quadratics, yes, if you keep getting X all the way to infinity, Y is going to keep increasing eventually going to go past the minimum point, and then you're going to be onto this increasing- so this is the minimum point that you're going past, here.

And then X is increasing, Y is increasing.

But actually beforehand it doesn't.

So that's why I'm going to put it as sometimes.

If you put it as never, I think you can maybe justify that because it's not always the case throughout the whole graph.

Thank you so much for all your hard work today.

I hope you've enjoyed the lesson.

Hopefully you're a bit more confident with sketching quadratics.

What are the two things you definitely need to ask yourself when sketching quadratics? Fantastic.

If it's a positive or negative quadratic, and then? What the Y intercept is.

There will always be a Y intercept, we found that out, and we can find that out by doing what? Excellent, substituting X equals zero into our equation.

Please do the quiz, it's an excellent way for you to check what you know and even do a bit more learning by using the feedback.

And I hope you have a wonderful day.

Bye!.