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WOMAN: In this interactive we will
again look at graphs of parabolas,

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this time, however, graphs where the
equation is given in general form,

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that is, y = x squared + bx + c,

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and we will consider the effects
of 'b' and 'c' on the graph.

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We will do this, though,
using the discriminant

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and also looking at a graph
that relates to the discriminant

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to help us explore this.

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So before we go any further
with the interactive,

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we'll just have a little look
at the algebra behind it

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and what's actually going on
in the interactive.

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So we're looking at equations
of the form y = x squared + bx + c

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and we know that the discriminant
could be useful

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in terms of telling us
about the number of x-intercepts

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for this equation.

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So discriminants are calculated using
the formula b squared minus 4ac,

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and in this case a is 1

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so the discriminant
for this particular quadratic

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is b squared minus 4c.

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So the discriminant's useful in terms
of the number of x-intercepts.

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So we know that
y = x squared + bx + c

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would have just one x-intercept
when its discriminant is equal to 0,

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and we can rearrange that little
equation we get there

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and say that that's when ...

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... any time that c is equal
to one-quarter b squared,

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our parabola y = x squared + bx + c
will have just one x-intercept.

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So over on the right here

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we're looking at a graph
of the discriminant,

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so we're looking at b
versus c here.

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So we're looking at the relationship
c = one-quarter b squared

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and what we know is that
any point on this red line here

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gives b and c values

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which will give the original
quadratic one x-intercept

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because they would be when
the discriminant is equal to 0.

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So for example, a point
on this line is (1, one-quarter),

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when b is 1, c is a quarter,

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or (2, 1), when b is 2, c is 1.

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So taking that,
what we're seeing here, then,

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is, well, if b is 2 and c is 1,

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so we get ... the original quadratic
is y = x squared + 2x + 1,

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that would be a quadratic
with just one x-intercept.

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And we could say the same thing for
every single point on that red line

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which gives us b and c values,

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which then make
that original parabola

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a parabola with just one x-intercept.

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And then we can think
about the other situations here.

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So we know that
we get two x-intercepts

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when the discriminant is positive,

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which, rearranging, is when c
is less than one-quarter b squared,

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so we're talking
about that entire region

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under the line y = a quarter
b squared shaded in green.

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So any point in that green region
should give us b and c values

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which make the original parabola
have two x-intercepts.

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So, for example,
a point in that green region

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is the point (4, 1),

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so when b is 4, c is 1
is in that green region.

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So that would mean that the parabola

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y = x squared + 4x + 1

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is a parabola with two x-intercepts.

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And we can look
at the last case also.

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We know we get no x-intercepts
when the discriminant is negative

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and that's when c is greater
than a quarter b squared.

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So any points in that
blue region there,

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for example, the point (2, 4),

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which would mean
b is 2 when c is 4

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so would give the parabola
y = x squared + 2x + 4,

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that would be a parabola
with no x-intercepts.

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And so this little graph here
on the right

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gives us all the different
relationships between b and c

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that gives us all the different
combinations of x-intercepts,

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that is, one, two or no x-intercepts.

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So if we return now
to the interactive,

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we can see how that's
actually working over there.

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So here what we're seeing is -
the graph at the top

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is the graph
of y = x squared + bx + c

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and the graph down here
in the bottom left

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is the little graph we were
just having a look at before,

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that graph to do
with the discriminant

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and the relationship
between b and c.

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So this line is the line
c = a quarter b squared.

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Any point on that line should give us
just one x-intercept.

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Any points below that line, in that
space, as you'll see written there,

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is when the discriminant is positive

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and hence is when we'll
get two x-intercepts,

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and any points above that line
will make the discriminant negative

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and hence will give us
no x-intercepts.

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So at the moment you'll see
we're looking at a point on the line,

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so when b is 0 and c is 0.

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That's on the line
c = a quarter b squared,

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and therefore that's
a combination of b and c

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that is giving our original graph,
x squared + bx + c,

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just one x-intercept.

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And so we should be able
to explore ...

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If we click anywhere
along this line -

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and we're not going to be able
to be 100% accurate here -

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but around about on the line

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should give us the original parabola
with about one x-intercept.

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So you'll see
I'm slightly off the line

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and the graph's sitting
slightly above or slightly below.

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But generally,
you're getting the sense

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that any pair of b and c values
that actually sit on this line

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c = a quarter b squared

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gives us a parabola which
has just the one x-intercept.

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And then looking in the region
below that,

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anywhere down underneath
this parabola

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is when the discriminant is positive,

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and hence we get parabolas
with two x-intercepts,

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b and c values that give the
original parabola two x-intercepts.

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And also having a look at the region

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above the line
c = a quarter b squared,

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we are seeing up here we get
combinations of b and c

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that make the discriminant negative

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and therefore make our original graph
have no x-intercepts.