Text description - screencast for interactive 2

[In this video, two electronic x and y graphs are used to show how equations can be presented in graph form. In the left-hand graph, an adjustable pointer allows individual values in the equations to be changed. As the values in the equations are changed, the parabola in the right-hand graph moves around the graph. In the left-hand graph, the section inside the parabola is shaded. The narrator describes the changes in the graphs.]

NARRATOR: In this interactive we will again look at graphs of parabolas, this time, however, graphs where the equation is given in general form, that is, y equals x squared plus bx plus c, and we will consider the effects of 'b' and 'c' on the graph. We will do this, though, using the discriminant and also looking at a graph that relates to the discriminant to help us explore this.
So before we go any further with the interactive, we'll just have a little look at the algebra behind it and what's actually going on in the interactive. So we're looking at equations of the form y equals x squared plus bx plus c and we know that the discriminant could be useful in terms of telling us about the number of x-intercepts for this equation. So discriminants are calculated using the formula b squared minus 4ac, and in this case a is 1 so the discriminant for this particular quadratic is b squared minus 4c. So the discriminant's useful in terms of the number of x-intercepts.
So we know that y equals x squared plus bx plus c would have just one x-intercept when its discriminant is equal to 0, and we can rearrange that little equation we get there and say that that's when… any time that c is equal to one-quarter b squared, our parabola y equals x squared plus bx plus c will have just one x-intercept.

[On an x and y graph, the x-axis is labelled b, and the y-axis is labelled c. Above the y-axis, a parabola passes through the point (0, 0). The parabola is marked c equals one-quarter b squared.]

NARRATOR: So over on the right here we're looking at a graph of the discriminant, so we're looking at b versus c here. So we're looking at the relationship c equals one-quarter b squared and what we know is that any point on this red line here gives b and c values which will give the original quadratic one x-intercept because they would be when the discriminant is equal to 0. So for example, a point on this line is (1, one-quarter), when b is 1, c is a quarter, or (2, 1), when b is 2, c is 1.
So taking that, what we're seeing here, then, is, well, if b is 2 and c is 1, so we get…the original quadratic is y equals x squared plus 2x plus 1, that would be a quadratic with just one x-intercept. And we could say the same thing for every single point on that red line which gives us b and c values, which then make that original parabola a parabola with just one x-intercept.

[The area below the parabola is shaded green.]

NARRATOR: And then we can think about the other situations here. So we know that we get two x-intercepts when the discriminant is positive, which, rearranging, is when c is less than one-quarter b squared, so we're talking about that entire region under the line y equals a quarter b squared shaded in green. So any point in that green region should give us b and c values which make the original parabola have two x-intercepts. So, for example, a point in that green region is the point (4, 1), so when b is 4, c is 1 is in that green region. So that would mean that the parabola y equals x squared plus 4x plus 1 is a parabola with two x-intercepts.

[The area inside the parabola is shaded blue.]

NARRATOR: And we can look at the last case also. We know we get no x-intercepts when the discriminant is negative and that's when c is greater than a quarter b squared. So any points in that blue region there, for example, the point (2, 4), which would mean b is 2 when c is 4 so would give the parabola y equals x squared plus 2x plus 4, that would be a parabola with no x-intercepts. And so this little graph here on the right gives us all the different relationships between b and c that gives us all the different combinations of x-intercepts, that is, one, two or no x-intercepts.

So if we return now to the interactive, we can see how that's actually working over there. So here what we're seeing is - the graph at the top is the graph of y equals x squared plus bx plus c and the graph down here in the bottom left is the little graph we were just having a look at before, that graph to do with the discriminant and the relationship between b and c. So this line is the line c equals a quarter b squared. Any point on that line should give us just one x-intercept. Any points below that line, in that space, as you'll see written there, is when the discriminant is positive and hence is when we'll get two x-intercepts, and any points above that line will make the discriminant negative and hence will give us no x-intercepts. So at the moment you'll see we're looking at a point on the line, so when b is 0 and c is 0. That's on the line c equals a quarter b squared, and therefore that's a combination of b and c that is giving our original graph, x squared plus bx plus c, just one x-intercept. And so we should be able to explore… If we click anywhere along this line - and we're not going to be able to be 100% accurate here - but around about on the line should give us the original parabola with about one x-intercept. So you'll see I'm slightly off the line and the graph's sitting slightly above or slightly below. But generally, you're getting the sense that any pair of b and c values that actually sit on this line c equals a quarter b squared gives us a parabola which has just the one x-intercept.

And then looking in the region below that, anywhere down underneath this parabola is when the discriminant is positive, and hence we get parabolas with two x-intercepts, b and c values that give the original parabola two x-intercepts. And also having a look at the region above the line c equals a quarter b squared, we are seeing up here we get combinations of b and c that make the discriminant negative and therefore make our original graph have no x-intercepts.