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Hello, I'm Mr. Coward, and welcome to today's lesson on Sketching Quadratic Graphs.

All you need for today's lesson is a pen and paper, or something to write on and with.

If you can, please take a moment to clear any distractions, including turning off your notifications, that would be great.

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

Okay, when you're ready, let's begin.

Okay, so time to the trainer's task.

I'd like you to solve these two equations, and then try and work out which graph shows which one, and explain how you know.

Can you find three different reasons? So pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Now hopefully you've solved these two correctly.

And so for this one we should have had, in factorised form, we should have had that.

Which means our solutions were this.

Okay? And this one, you should have factorise like this.

So that means our solution was just that.

Now, which one did you think was which? Well, you might have recalled from when we are solving equations graphically, and when we were solving quadratic simultaneous equations, that that line, is like saying when does y equals zero? So the solutions to this tells us that it's the graph of y equals x squared.

So yeah, y equals x squared minus four x plus three, crosses, y equals zero, at one and three.

So this graph crosses why equals zero at one and three.

So we're told this from the graph.

So that means that this graph here equals y equals x squared minus four x plus three.

So these solutions, which we have a fancy word for, and we call them roots.

These solutions are called roots.

I'm going to use that word a lot, so I want you to get used to it, roots.

So these roots of our equation tell us where the graph crosses y equals zero, which is the x axis.

So this one tells us, well, it doesn't tell us it crosses it because there's only one.

If there's only one root, it just touches it.

So you could say, one of the reasons you could say that the red line, or you know that this red line is y equals x squared minus five x plus three because it has roots or solutions at one and three.

Or you could say because it crosses twice, whereas the other one only crosses once, and that has one route or solution.

Now, what could another reason be? Well, this one crosses at plus three, and this one crosses at plus nine.

Did you know is that? What else? Well, you could have substituted in that point.

You could have substituted in one x equals two, into this one, so you get two squared four, minus four times two, eight, plus three.

So you've got four plus three minus eight, which gives us negative one, which gives us that point there.

And you could have done that for a whole manner of parts.

So there's actually loads of reasons or loads of different ways in which you can work out which graph which.

So just to confirm this red one is y equals x squared minus four x plus three, and this one equals this.

All right, now there's four main features needed to sketch a quadratic graph.

The general shape.

So we can have two shapes.

We can have this kind of shape for a quadratic graph, all this kind of shape.

This one is when it's positive, and this one is when it's a negative x squared two.

So when the coefficient of x squared is positive, it looks like that, when the coefficient of x squared is negative, it looks like that.

But we'll explain more as we go through this lesson.

So don't worry if you don't fully understand yet.

Okay, here are the roots.

And that's just where we solve the graph of y equals zero.

And then that tells us average.

Here's the y intercept, where it crosses the y axis, where it crosses or intercept the y axis.

Also note that the roots are where the graph crosses the x axis.

So they can sometimes be called the x intercepts.

And the final point is the vertex or turning point.

This is either the minimum or maximum point of a graph.

So the highest point or the lowest point.

In this case, it's a minimum, it's the lowest point.

Where if it was a negative graph, our turning point would be a maximum.

And we'll talk we'll talk more about these three in today's lesson, and we're not really going to touch on that too much.

Okay, so the roots of a quadratic tells us where the graph intersects the x axis.

So if I solve this, we can see that the roots of this graph, which is y equals x squared plus two x minus three, y equals that thing, has roots negative three and one, which is the same as the x values for the solution of the equation when the equation is equal to zero.

So when that's like when y is equal to zero, And that makes sense, because on that line, our y coordinate is equal to zero.

Okay, so we're going to find the roots of the following graphs.

So, first thing we do, is we set up an equation of the graph equal to zero.

So I'm setting that equal to zero.

Now I factorise.

So then remember that this bracket is equal to zero, or this bracket is equal to zero, so we get this.

So your turn.

So pause the video and have a go, find the roots of the following graph.

Pause in three, two, one.

Okay, welcome back.

Hopefully you set equal to zero first actually.

And then you factorised which gives you that equals zero all that And then we get our two roots.

Okay, so those are the roots of the equation.

Where the graph crosses the x axis.

So the x intercepts or where the graph crosses the line y equals zero.

Okay, so sometimes we can have no roots.

Sometimes the graphs don't cross the x axis.

Sometimes we can have one root, and if you factorise, this, you would get that.

So we our solutions would be x equals negative three.

This one, you would not be able to factorise, and so we'd say this one has no real roots.

It actually has imaginary roots.

This one, this one would factorise into that, so you'd get your roots, x equals negative four and x equals negative two.

And you can see those are the roots.

And sometimes this one, well, we can't actually factorise this one, however, it does have roots but these are just not integer roots.

So we'd have to find the roots of this using a different method and which we're not going to cover in this unit.

Okay, the y intercept.

The y intercept of a quadratic graph tells us where the graph intersects the y axis.

Y intercept, where it crosses the y axis.

Now, can you see, how this is related to the y intercept? This point here? Well, it's just this path.

But why is it that path? So what is the x coordinate of every point on this line? Every point on the y axis has an x coordinate of zero.

So, to find the y coordinate, we substitute in x equals zero.

So we have zero squared plus two times zero, minus three.

Zero squared plus zero.

Zero plus zero minus three, is just equal to negative three.

So we get y equals negative three.

So that, is how we get our y intercept.

We substitute in zero, and actually just gives us this last part here.

Okay, so find the y intercepts of the following.

My turn.

y equals zero squared, plus four times zero minus 21.

So zero squared is zero plus five times zero, negative 21.

Okay, so your turn, pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Hopefully you've got this, zero squared plus five times zero plus six.

So you just get the six.

The positive six.

So it's just this part.

And why is it just that part? Because when it crosses the y axis, the x coordinate is equal to zero.

So we substitute in zero and we just get the constant term.

Okay, now the coefficient.

So remember, that's the thing in front of the x squared.

So if the coefficient is a, the coefficient of x squared tells us the shape.

If a is greater than zero, we have this u shape.

Or some people think of it as a smile.

So if a is positive, we get a smile.

And so people think of this as a frown, or an n shape.

So if it is negative, we get a frown or an n shape.

Okay, so I want you to identify which quadratics have a U shape and which have an n shape.

So pause the video and have go.

Pause in three, two, one.

Okay, welcome back.

Now hopefully you got that this one was positive, so this one was a u and this one was negative so we have an n shape positive, negative.

'cause it's the thing that's with the x squared.

It's not the first thing, it's the thing that's with the x squared, which is negative or half, positive, And where's our x squared? Here's our x squared there, so it is positive.

Really well done if you've got them correct.

If these two were it was in a slightly different order tricked you a little bit, don't be fooled next time.

So, how to find a route is negative? Well, what we can do is we can think of it in two different ways.

So the first way that we can think of it, is like this.

So, when is this equal to zero? So when is y equal to zero? And then we do our times by negative one on both sides.

Well, zero times by negative one is just zero.

So then this side becomes x squared plus x minus 20.

So all the signs change, so it goes from being negative x to positive x, negative one x to positive one x and positive 20 to negative 20.

Then we can do this factorise this x plus five, x minus four and that gives us our roots.

Or the second way that you can think about it and some people like to factor out the negative.

So they do negative of x squared plus x minus 20.

And then they factorise it and find the roots.

Both are kind of the same.

I slightly prefer this way, but it doesn't really matter that much to be honest.

So if you want to do that first and then say that is equal to zero, then factorise so we have a negative still outside, X plus five, x minus four is equal to zero.

And then what you can do with the negative is you can bring it back in, and I'll bring it back into this one.

It doesn't actually matter which one you bring it back into, and then that's what's the sign of this one.

So you'd get negative x plus four equals zero, and you'd get x plus five equals zero, or x.

Negative x plus four, equals zero.

Then you could add x to both sides, and you'd get four equals x or x equals five.

So they're the two ways you can do it.

I slightly prefer this way, but I just wanted to show you the other way, just in case you like that way better.

So I would like you to have a go at that.

So pause the video and have a go.

Pause in three, two, one.

Okay, welcome back.

Now I've just cleared my screen 'cause there's a lot of stuff in the way.

Hopefully you said this equal to zero, then times both sides by negative one, and then factorise.

If you did it the other way, that is completely fine, you'll get the same answers.

and it's just down to preference really.

And so that would be negative 10 plus two, then we'd get x equals 10 or x equals negative two.

So really well done if you've got that correct.

Okay, so now it is time for the independent task.

So I would like you to pause the video complete a task and resume once you've finished.

Okay, so here are my answers.

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

Okay, and now it's time for the Explore task.

So a graph has a line of symmetry, which helps find the turning points.

What do you notice about the x coordinates of the turning points and the roots? So remember, the turning point would be this point here.

So, what do you notice about the turning point of the x quadrant of the turning point and the roots? Can you use this relationship to find the x quarter of the turning points of these four graphs here? How could you now find the y coordinate? So that's something to think about.

Okay, just before you pause the video, I want to let you know that I have got a hint.

But I want you to try this first.

So pause the video to complete your task.

Come back for the hint if you struggle in, and if you haven't struggled and you think you'd got the task, come back afterwards for my answers.

Okay, so my hint is here.

Well, this is my line of symmetry down there, okay? Not greatly drawn, I'm sorry about that.

But can you see, what do you notice about this distance here, and this distance there? Two, two.

They're the same.

And in fact, any two points on that line are the same distance away from the line of symmetry.

So we can say, whenever we have two points, we know that this is going to be this point here, the x coordinate of that point, is going to be halfway between those two points.

So that's your hint for this part, and then you hit for this part, how could you find the y quarter? You're going to have to use substitution for that.

So if you haven't had a go already, or if you got super far, have gone now and try and do it and come back once you're finished.

Okay, so here are my answers.

And how could you find the y coordinate? You could substitute in the values of the x coordinate.

I didn't actually ask you to do it, but I can just do it for one of them here.

So for this one, you would have had x squared, when x is three, so you'd have nine, three times three, nine, minus six times three, plus eight.

So nine minus 18 plus eight, which would give us negative one, I believe.

Nine plus eight is 17, 17 minus 18, is negative one.

So for this graph, the turning point would be three negative one.

Okay, so that is all for this lesson.

Thank you very much for all your hard work, I look forward to seeing you next time.

Thank you.