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WOMAN: In this interactive,
we're looking at graphs of the form

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y = a(x – h) cubed + k.

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So here we have three constants,
'a', 'h' and 'k',

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which are going to affect the graph
of y = x cubed in different ways.

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So at the moment
we have 'a' set to one,

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'h' set to zero and 'k' set to zero,

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so we are, in fact, looking
at the graph of y = x cubed.

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So let's first explore what happens
as we alter the value of 'a'.

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So as we increase the value of 'a'

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we see the graph
appearing to get thinner,

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and, in fact, the transformation
that's occurring

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is a dilation away from the x-axis,

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so it's the graph being stretched
away from the x-axis

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and hence appearing thinner
as well as taller.

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So resetting 'a' back to one,
there's our graph of y = x cubed.

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Now if we look at
a fractional value of 'a',

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so 'a' equal to a half,

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we're seeing still a dilation
from the x-axis,

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but a dilation from the x-axis
by a factor of a half

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causes the graph to squash down
towards the x-axis

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and hence appear fatter.

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If we make 'a' zero we don't
have a cubic function at all

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but just the linear graph y = 0.

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And now, looking at
negative value of 'a',

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we see the graph reflected
in the x-axis

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and we're also getting
that dilation effect

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to do with the actual magnitude
or size of 'a'.

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So for example, at the moment we're
looking at the graph of y = –3x cubed

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and this is the graph of y = x cubed
after both a reflection in the x-axis

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and a dilation from the x-axis
by a factor of three.

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So 'a' can impact
two transformations,

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so reflections in the x-axis
and dilations from the x-axis.

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Let's now consider 'h'.

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So 'h' is the term

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inside of the cubed bracket,
so (x – h) all cubed. 

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So as we make 'h' larger, we see
the graph moving to the right.

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So here, for example,
is the graph of (x – 3) all cubed

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and this graph has moved
to the right by three.

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(x – 4) all cubed and we've moved
to the right by four.

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If we make 'h' negative ...

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Here we're looking at
(x – –3) all cubed,

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so that is (x + 3) all cubed,
which has gone to the left by three.

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Here we're looking at (x – –5) all
cubed, so hence (x + 5) all cubed,

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and that graph has gone
to the left by five.

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So setting 'h' back to zero.
Now let's explore 'k'.

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So as we increase the value of 'k'

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we see the graph moving upwards.

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At the moment we're looking
at the graph of y = x cubed + 2

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and this is the graph of y = x cubed
translated up by two.

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As we reduce the size of 'k'...

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We're now looking at the graph
of y = x cubed – 3

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and we see that this is the graph of
y = x cubed translated down by three.

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So we note that 'h' and 'k'
both affect translations.

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'h' affects horizontal translations

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and we need to be a little
cautious about 'h'.

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For example, (x + 2) all cubed
has gone to the left by two,

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which seems a little
counterintuitive,

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and (x – 2) all cubed
goes to the right by two.

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'K' affects horizontal - sorry -
vertical translations

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and they are less counterintuitive

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in that x cubed + 2
in fact goes up by two

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and x cubed – 2 does indeed
go down by two.

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So 'a' is dilations
and reflections of the graph,

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'h' and 'k' translations.