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WOMAN: In this interactive,
we explore graphs of parabolas,

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parabolas where the equation
is given in the form

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y = a(x minus h) all squared + k.

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This form is referred to
as turning point form.

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We're currently looking
at the graph of y = x squared.

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That is, where an equation
in turning point form

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has an a value of 1, an h value of 0
and a k value of 0.

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This interactive allows us

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to explore the effects of a, h and k
on the parabola.

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Let's first consider
the effects of a.

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a is currently set to 1

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and we'll first consider what happens
as we increase the value of a.

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And we see as a increases, we're
getting some sort of a dilation

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and, in fact, the dilation
that is occurring

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is a dilation away from the x-axis.

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So the graph we're seeing at the
moment is the graph of y = 3x squared

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and that's the graph of y = x squared

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after a dilation of a factor of 3
from the x-axis.

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That is, the point that was at (1, 1)
is now at (1, 3).

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It's been stretched by a factor of 3
away from the x-axis.

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So a certainly affects
the dilation of the parabola,

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and we note that as we've decreased
the value of a

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and I've stumbled across a = 0,
we're seeing nothing.

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Well, in fact, we're seeing
the equation y = 0,

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which is a horizontal line
along the x-axis.

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But as we make a negative
we see the parabola reappear

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and we're now seeing a parabola
that is upside down.

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So negative values of a
reflect the graph in the x-axis.

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We're still seeing

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as we make the magnitude of a larger,
the graph is being dilated,

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so the magnitude of a

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is a dilation from the x-axis
by a factor of a,

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and the sine of a

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is to do with whether the parabola
is up the right way or upside down,

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so a negative a value
causes a reflection in the x-axis.

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Let's now consider the effects of h.
h is currently set at 0.

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So we'll look at what happens
as we increase the value of h,

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and we see that as we increase h,
the graph is moving to the right.

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So here, where h is equal to 6,

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that is the equation
is (x minus 6) all squared,

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the graph has been moved
to the right by 6.

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Decreasing the value of h,

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it would be no surprise to see
the graph moves instead to the left.

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So here, for example, where h
is equal to negative 4,

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that is the equation we're seeing
is y = (x + 4) all squared,

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the graph has been moved to the left
by a factor ... by 4.

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And so the key thing to note with h

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is that when looking
at the equation of the graph

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if in that bracket you're seeing
(x + something) all squared,

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the graph has moved to the left
by that factor.

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If you're, in that bracket, seeing
(x minus something) all squared,

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the graph has, in fact,
moved to the right by that factor,

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as is the case here, right by 3.

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And lastly, let's consider
the effects of k,

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and at the moment k is set to 0

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and we see that as we increase
the value of k,

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the graph, in fact, moves up.

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So here where k is 8, and hence we
have the equation y = x squared + 8,

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we've seen the graph move up by 8,

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and as we decrease the value of k,
making it negative,

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we see the graph move down.

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So here where the value
of k is negative 5,

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we're seeing the equation
y = x squared minus 5

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and the graph, the parabola,
has been translated down by 5.

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And so, obviously, we could put all
of these transformations together,

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the dilations and reflections
caused by a

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and the translations caused
by h and k,

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to create any kind of parabola
in any kind of position.

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Now, we mentioned previously that
this form of a quadratic equation

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is called turning point form

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and that's because, as we'll
see here, for example,

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when we make h 4 and k 5
and hence we have the equation

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y = (x minus 4) all squared + 5,

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we see a direct correlation between
those numbers, those translations,

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and the turning point of this graph.

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Obviously, this is the graph
of y = x squared

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after a translation to the right by 4
and up by 5,

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and that would therefore
make the turning point

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have coordinates (4, 5),

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and we should be able
to read those coordinates

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directly out of the equation,

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looking at the h value
and the k value.

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So turning point
of a graph in this form

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is, in fact, at (h, k).