Lesson video

Hello, I'm Mr. Coward, and welcome to today's lesson on sketching quadratic equations, part two.

For today's lesson, you'll need a pen and paper or something to write on and with and a ruler or some kind of straight edge.

If you can, please take a moment to clear away any distractions, including turning off any notifications.

And if you can, try and find a quiet space to work where you won't be disturbed.

Okay.

When you're ready, let's begin.

Okay.

So time for the Try this task.

I'd like you to pause the video and have a go.

Pause in three, two, one.

Okay.

So hopefully you've had a go.

Now, my roots.

Well, that is when x equals negative three and that means x equals negative one.

Sorry, positive one.

And because those roots are solutions to the equation when my graph, my equation of my graph is equal to zero.

So what I can do now is work backwards and see when this is true.

Well, that and that can be combined to this statement.

So if that equals zero or that equals zero, then this thing equals zero.

Now I can expand it.

Simplify.

Okay.

Now I can put y equals this, but there's one thing I need to consider beforehand.

Is my graph positive or is my graph negative? Well, if my graph was positive, I'd get this.

And I can see that my graph is positive.

Okay.

A positive coefficient of x.

However, if it was negative, I would have the same roots.

It looks something like that.

Okay.

It'd be a reflection of my graph in this line here.

Okay.

A reflection of my graph in the x-axis.

Now, we can see that that's not right for a few reasons.

Obviously, because the shape is not the same as our graph.

And if it was the negative version, we'd have times that by negative one.

Times that by negative one, times that by negative one.

We'd have that and it would cross at plus three.

However, you must be careful.

You must be careful because if I had expanded this and that was a plus three, well, then I could see that that wasn't right.

And it's likely that it would have been a negative version of my graph.

So I just want you to watch out for that, okay? Watch out for working backwards because we need to make sure if our graph is positive or negative and that we have the right equation there.

Okay.

So sketch the following graphs.

So we've got to get onto some sketching now.

Now when I'm sketching, I always like to draw in my graph part first and then the axes.

So the shape, we've got a U shape, so we've got positives here.

So it's going to be something like this.

Okay, so my shape's done first.

Now, where does it cross the y-axis? It crosses at zero, two.

That's my way intersect.

And what are my roots? They are my roots.

Negative one and negative two.

So those are both negative.

So here's my x-axis here.

That point there is negative two.

That point there is negative one and now I'm going to have my y-axis.

And that is going to be my y-intercept, which is at the point zero, two.

Okay, so my graph should look something like that.

Okay.

So for your turn, I'd like you to pause the video and try and sketch this graph.

So remember, draw the shape first, then do either, it doesn't really matter what you do first.

Your roots or your y-intercept.

I'd probably do roots first, actually.

And then your y-intercept last.

So pause the video and have a go! Pause in three, two, one.

Okay, welcome back.

Now, we've got a U shape.

Okay.

Hopefully you've done that.

And then our roots, we've got one at one and we've got one at negative two.

This is going to be negative two, and this is going to be one.

And we know that our y-axis is going to be closer to the one than the two.

So it's going to be something like that.

Probably not very accurately drawn over there, but I mean, it doesn't, the key points are getting our key points.

I mean, I would have preferred it if this had gone slightly more like that up there.

And that was negative one there, but it doesn't really matter too much.

And this point here would be zero, negative two.

So my y-intercept would be zero, negative two.

So it's just to get a general idea and try and make it accurate, try and have, so for instance, that line should have been about two-thirds there and a third there and to be honest, these should have been, this graph should have been a bit flatter like that, a bit flatter like that.

And then I should have maybe moved the axis over there, but it doesn't really matter that much.

As long as you kind of get in these points and it look roughly sensible, then that is fine.

Okay.

So sketch the following.

Well, we have a lot less information here.

We need to find our roots first.

So to find our roots, we're going to factorise and set equal to zero.

So yeah, x squared minus five x minus six equals zero.

Factorise.

Sorry, x minus six, x plus one.

So then we get our roots.

What is our y-intercept? Our y-intercept is zero, negative six and that's positive.

So we're going to have a U shape.

So we get all the information from the graph now, and then we're drawing.

So it crosses at six and negative one.

We have a U shape.

So this, you would draw, this is my axes.

Now this here is the point six.

This is a point negative one.

So my y-axis is going to be much closer to that side.

And my turning point, not that you need it per se, but it can help, is halfway in-between.

So that's halfway in-between.

That's obviously less than halfway in-between, so the bottom of my graph's not going to be on this side.

And then that line will be negative six, okay? So that's at the point negative six.

So it's a rough sketch, draw our shape.

We mark on the roots, we add in our y-axis and we mark on our y-intercept.

Pay attention to where the turning point is 'cause that's halfway in-between our roots and that's about it really.

So you find your roots, you find your y-intercept and you find the shape.

So pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Hopefully if you've got the shape, it's positive.

Hopefully you got the, oops, sorry.

Hopefully you got the y-intercept is, that is an awful arrow, zero, negative six, so the same as mine.

And if you've factorised this, you should've got this, I think.

Yeah, you should have got that! Which means that it crosses at x equals negative three or x equals positive two.

So drawing my graph on my shape.

So negative three and positive two.

So my axis is going to be closer to this side.

Bit of a wonky axis, I am sorry.

It's quite difficult drawing with this pen.

And that point there is going to be negative six.

Okay.

Hopefully yours look something similar to mine.

Maybe with a less wonky axis.

Maybe a ruler is probably sensible for the axis.

Okay.

Sketch the following.

So we do the same thing.

We factorise, to find the roots.

So that means x plus three equals zero, or x minus three equals zero, because we're not asking for y anymore.

We're asking when that thing is equal to zero.

So that means that bracket's equal to zero or that bracket's equal to zero.

x equals positive three.

Okay.

So they are my two roots.

My y-intercept is zero, negative nine and it is a positive shape.

Okay.

So let's draw that in.

There's my shape.

Ah, now you see how that, and that, well they're equidistant, they're both three away from the y-axis.

So the bottom of my graph is going to be that point there on my y-axis.

The bottom line of my graph is going to be negative nine because our turning point is halfway in-between negative three and three.

Okay, so this one.

Pause the video and have a go.

Pause in three, two, one.

Okay, hopefully you found that the roots were this.

The shape is this and the y-intercept is zero, negative four.

And then hopefully you had beautiful sketch.

Like this clearly is.

I think its not too bad to be honest.

Quite happy with that.

Negative two and two, and that point there is the.

0, negative four.

Okay? So really well done if you got that.

Okay, well, what's different now? How's this changed? Well, we just factorise it differently because they've both got an x in common and there's no constant term, we're just going to factorise that into a single bracket.

So that is equal to zero.

So that means that is equal to zero.

Well, that is equal to zero! So now, when we're drawing that curve, we know that one of these lines is going to be on the y-axis.

So one of these points and it's going to be this point here, is going to be on the y-axis.

'Cause that's when x equals zero.

That's when x equals nine and that's going to be, so that's our y-intercept.

We have here, I should have mentioned that, I'm sorry.

Our y-intercept is zero, zero.

It's the origin.

Okay? Because there is no constant at the end.

So because there is no constant at the end, the y-intercept is zero, zero, which is also a root.

So it's actually quite nice when we don't have a constant term on the end.

Could you work out this point from the symmetry? Or the x-coordinate? Well, that would be about 4.

5.

You don't need that for now, but I just, you know, that's interesting.

Okay.

So have a go at the Your turn one.

Pause the video and have a go in three, two, one.

Okay.

Welcome back.

Hopefully you factorised this like this, which means our roots are this.

So then we sketch that and we've got x equals zero there, x equals negative four there and our y-intercept is the origin.

Okay.

So really well done.

Okay.

So this is going to be the last one, and this is going to be negative this time.

So it's going to be this shape.

Okay.

My y-intercept is going to be zero plus three.

Now, how do we find the roots again? Well, we've got this is equal to zero.

So then what do we know? Well, we can times both sides by negative one.

That changes the signs of each of these terms. Then we factorise and that gives us our roots.

Negative three and positive one.

Okay, so we need to draw it now.

Now it is not this shape anymore.

It is this shape.

So we've got to be really careful that we don't draw in the wrong shape.

So we've got a negative quadratic.

Now it crosses at negative three and negative one.

So draw in on an axis.

Here we've got negative three.

I'm sorry, here we got positive one.

Positive one is further along the number line than negative three.

And here we've got negative three.

So where is my axis going to be? Well, it's going to be closer to this one.

It's going to be three-quarters of the way along or one-quarter going that way.

Like that, and then that point there is going to be positive three.

Zero, positive three, sorry.

Okay.

So that's that quadrant, zero, positive three.

So you could see this time it's similar, but we've just got to remember that the shape is different and that to find our roots, we need to times everything on both sides by negative one.

So pause the video and have a go.

Pause in three, two, one.

Okay, so hopefully you have found that the y-intercept is zero, negative four, and you have found that it is this shape, and you have also found the roots.

Just times by my negative one.

Factorise.

So x equals four or x equals one.

So we're going to have this.

Something like that.

And it crosses at four and one.

This is going to be one and that's going to be four 'cause four is further along the number line.

And I'm going to, I'm going to go a little bit more that way.

I might go a little bit more that way.

So if I was using a rubber here, that would be a bit easier.

Just the reason why I did that and that made it go a bit more out that way is just so I can fit on my y-intercept, which is going to be there.

So that's my y-intercept of negative four.

And when you sketch in them, it doesn't matter if you write it like that or if you write it as a coordinate.

Okay.

So it should look something like that.

Going nice and wide.

Crossing at one, four.

Crossing at negative four and with a turning point in-between, halfway in-between them two.

Okay.

So now it is time for the independent task.

So pause the video to complete your task and resume once you've finished.

Okay.

So here are my answers.

You may need to pause the video to mark your work.

So now it's time for the explore task.

So A, B is the turning point, and you've got to find all these missing points.

So A, B, C, and D.

Now I have a hint, but have a go first.

So pause the video to complete your task and resume once you've finished.

So time for my hint.

So my first hint is that this line is halfway between that and that.

Those are the same distances.

So it's halfway in-between.

And that and that.

That line is halfway in-between.

So that distance is the same as that distance.

And my other hint is for this point here, and for this point, you may need to use substitution.

Now, how can we use substitution? We don't have the equation.

Well, you've got to work out what the equation is.

Okay, so here are my answers.

I'm going to talk through it.

So this line, my line of symmetry, is halfway between that and that.

So they are six apart.

So it is that distance there is going to be three and that distance there is going to be three.

So that will mean A is negative two.

Okay.

Now I've found A, I need to find B.

Well, how can I do that? So what about this? Let's work out what the graph is first.

So we've got x minus five.

Sorry, x equals negative five or x equals positive one.

So that means, sorry, x minus one equals zero.

Okay.

Now we're going to make our graph.

So we get y equals, expand this, x squared plus five x.

Now, how did I know that it wasn't negative? And I had to times it by minus one? Well, because this graph was that shape, which is positive.

So I didn't worry.

I kind of rushed that step, getting to there because I knew it wasn't going to be a negative.

So now I've found my equation of this.

So because I've got that equation, I can substitute in negative two.

So you get y equals negative two squared, plus four times negative two minus five.

So we get four minus eight minus five.

So four minus eight is negative four.

Negative four minus five is negative nine.

That means B is negative nine.

Now we found them, we can use substitution to find C.

So let's find C.

x equals negative six.

So we have negative six squared.

And I'll just write my answer.

36 plus four times negative six, which is negative 24 minus five.

So 36 minus 24 is 12.

12 minus five is seven.

So you should have seven there.

And that is going to be the same.

That's going to be seven.

And that is how many away is that? That's four away.

So that's four away in that direction.

So that will be negative two plus four.

So that will be two.

So really, that was quite a tricky task.

I am not going to lie.

So really well done if you got there.

Okay and that is all for this unit.

Thank you very much for all your hard work and really, really well done.

So if you'd like to, please ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

Thank you very much.

Take care.