WOMAN: In this interactive, we're looking at graphs of the form
y = a(x – h) cubed + k.
So here we have three constants, 'a', 'h' and 'k',
which are going to affect the graph of y = x cubed in different ways.
So at the moment we have 'a' set to one,
'h' set to zero and 'k' set to zero,
so we are, in fact, looking at the graph of y = x cubed.
So let's first explore what happens as we alter the value of 'a'.
So as we increase the value of 'a'
we see the graph appearing to get thinner,
and, in fact, the transformation that's occurring
is a dilation away from the x-axis,
so it's the graph being stretched away from the x-axis
and hence appearing thinner as well as taller.
So resetting 'a' back to one, there's our graph of y = x cubed.
Now if we look at a fractional value of 'a',
so 'a' equal to a half,
we're seeing still a dilation from the x-axis,
but a dilation from the x-axis by a factor of a half
causes the graph to squash down towards the x-axis
and hence appear fatter.
If we make 'a' zero we don't have a cubic function at all
but just the linear graph y = 0.
And now, looking at negative value of 'a',
we see the graph reflected in the x-axis
and we're also getting that dilation effect
to do with the actual magnitude or size of 'a'.
So for example, at the moment we're looking at the graph of y = –3x cubed
and this is the graph of y = x cubed after both a reflection in the x-axis
and a dilation from the x-axis by a factor of three.
So 'a' can impact two transformations,
so reflections in the x-axis and dilations from the x-axis.
Let's now consider 'h'.
So 'h' is the term
inside of the cubed bracket, so (x – h) all cubed.
So as we make 'h' larger, we see the graph moving to the right.
So here, for example, is the graph of (x – 3) all cubed
and this graph has moved to the right by three.
(x – 4) all cubed and we've moved to the right by four.
If we make 'h' negative ...
Here we're looking at (x – –3) all cubed,
so that is (x + 3) all cubed, which has gone to the left by three.
Here we're looking at (x – –5) all cubed, so hence (x + 5) all cubed,
and that graph has gone to the left by five.
So setting 'h' back to zero. Now let's explore 'k'.
So as we increase the value of 'k'
we see the graph moving upwards.
At the moment we're looking at the graph of y = x cubed + 2
and this is the graph of y = x cubed translated up by two.
As we reduce the size of 'k'...
We're now looking at the graph of y = x cubed – 3
and we see that this is the graph of y = x cubed translated down by three.
So we note that 'h' and 'k' both affect translations.
'h' affects horizontal translations
and we need to be a little cautious about 'h'.
For example, (x + 2) all cubed has gone to the left by two,
which seems a little counterintuitive,
and (x – 2) all cubed goes to the right by two.
'K' affects horizontal - sorry - vertical translations
and they are less counterintuitive
in that x cubed + 2 in fact goes up by two
and x cubed – 2 does indeed go down by two.
So 'a' is dilations and reflections of the graph,
'h' and 'k' translations.