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Skip to 0 minutes and 13 seconds So what we want to do now is we want to understand how the algebra of a parabola or quadratic relation interacts with the geometry. So we’ll be looking on the Cartesian plane, and we’re going to be thinking about equations of quadratic relations. What happens if we shift them around? We’re going to be looking at a very famous theorem of Descartes, which connects the geometry of a parabola with its zeroes. And we’re going to go back to the ancient Arabs, to a mathematician named Al-Khwarizmi who introduced this famous operation of completing the square, and explain how his completing the square operations really connected very naturally with geometry. So we’ve been talking about quadratic relations and how to visualise them.

Skip to 1 minute and 3 seconds So our basic quadratic relation is y equals kx squared for some constant k. And here are some examples of particular values. When k is 1, we’re talking about our standard parabola that we’ve already had a look at, y equals x squared. What happens if we change the k value? Well, then the shape of the parabola changes, but in a rather predictable way. So, for example, if we go to y equals 4x squared, then what happens is that, at any given value, we have to take the value of y equals x squared and simply multiply it by 4. So the net effect is to dilate the entire parabola in the vertical direction by a factor of four.

Skip to 1 minute and 46 seconds It’s a very simple geometrical transformation, it’s a dilation in the vertical direction. What if we introduce a negative value of k. So, for example, y equals minus 1/2 x squared? Well, the effect is similar, except that the negative sign here has an important effect of reversing the orientation of the parabola. So now it’s a problem opening downwards. The effect of the 1/2 is to widen the parabola. So instead of making it narrower, like the 4x squared, it now becomes a little bit bit wider.

Skip to 2 minutes and 24 seconds It turns out that the shape of all these parabolas is really essentially the same, that this parabola and this parabola are really just different versions of each other. So in other words, if we looked at the blue parabola and zoomed out a fair distance from it, it would look just like the green parabola. So this is a basic understanding, the basic quadratic relation. Now, however, we want to go a little bit further, and we want to investigate more general quadratic relations. We want to ask, what happens if we look at an expression of the form y equals a x squared plus bx plus c? This is a general quadratic relation.

Skip to 3 minutes and 8 seconds And beautifully it turns out that if we understand these ones, we can understand these. So it turns out that a general quadratic relation like this is obtained from one of these by a suitable translation in the x direction followed by a translation in the y direction. So we’re going to have a look at that. So let’s see what happens when we translate a parabola in the xy plane. Here is y equals 4x squared. So let’s take this parabola and translate it by 3. So the vertex, which is over here at 0,0, goes to the point 3,0. What we do to the equation is we take the equation, we replace the x with an x minus 3.

Skip to 3 minutes and 57 seconds So it becomes y equals 4 times x minus 3 squared. Let’s check that that actually works. If we plug in x equals 3, then, yes, we get 0 on the right-hand side, so y really is 0. So, yes, that really is the vertex. And now what about a shift in the vertical direction? If we shift it down by 1, that’s even easier. We just take the form y equals 4x minus 3 squared, and we subtract 1. The effect of that is to move each of the points down by 1 in the y direction. And, of course, the vertex here is the point 3 minus 1.

Skip to 4 minutes and 38 seconds And you could check if you plug in x equals 3, then, yes, y equals minus 1. So the lesson is that we can get at a general parabola

Skip to 4 minutes and 50 seconds if we have it in this form: some number, like the constant k, times x minus something, or perhaps plus something if it’s moving to the left, all squared, and then plus or minus something else. So once we have a parabola in this form, we can read off a lot about it. The 4 in front of the x minus 3 squared is telling us how steep or narrow the parabola is. The 3 in here and the minus 1 here are really the coordinates of the vertex. So we have very precise geometrical information about where this parabola is once it’s in this very special form.

Skip to 5 minutes and 32 seconds So a natural question is, suppose that we have a general quadratic equation, y equals a x squared plus bx plus c. First of all, can we put it in such a form? That would be a good thing to do because, if we can, we can immediately read off where the thing is. And how are we going to do that? The answer was given by an Arab mathematician about 1,000 years ago.

Zeroes, completing the square, and the quadratic formula I

Welcome back now to Week 4, as we put together a lot of the things you have learnt, and get into some deeper applications of quadratic relations and their algebra.

You have already seen that the basic linear proportionality \(\normalsize{y=x}\) can be rescaled into \(\normalsize{y=ax}\) and then translated to form the general line

\[{\normalsize y=ax+b }.\]

The quadratic situation is similar. The basic parabola \(\normalsize{y=x^2}\) can be rescaled into \(\normalsize{y=ax^2}\) and then translated so that the vertex is at the point \(\normalsize{[u,v]}\). This process forms the general parabola

\[\normalsize{y=a(x-u)^2 + v}.\]

This gives us a handle on more general quadratic relations, and also gives a strong visual orientation to the algebra of quadratic equations.

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