WOMAN: 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 = x squared + bx + 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 = x squared + bx + 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 = x squared + bx + 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 = x squared + bx + c will have just one x-intercept.
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 = 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 = x squared + 2x + 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.
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 = 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 = x squared + 4x + 1
is a parabola with two x-intercepts.
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 = x squared + 2x + 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 = x squared + bx + 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 = 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 = a quarter b squared,
and therefore that's a combination of b and c
that is giving our original graph, x squared + bx + 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 = 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 = 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.