WOMAN: In this interactive, we will explore graphs of quartic functions,
that is, quartic functions with equations of the form
y = ax to the power of four + bx cubed
+ cx squared + dx + e.
With five unknowns, 'a', 'b', 'c', 'd' and 'e',
there are numerous different combinations that we could create
and so we won't attempt to explore every possible combination
of 'a', 'b', 'c', 'd' and 'e',
but we will try to look at the different graph shapes
that we can obtain when graphing quartic functions.
So currently we have 'b', 'c', 'd' and 'e' set to zero
and 'a' set to one.
So we're looking at the graph of y = x to the power of 4,
the most basic quartic function,
and we see this is a slightly parabolic shape
but with a flatter base and steeper sides.
And we can alter this basic shape by simple dilation,
so that is by changing 'a',
making it steeper or flatter or reflecting it.
But this essentially forms one type of quartic function.
We'll note that it's a symmetric shape
and that's to do with the even power, obviously.
Similarly, we can create other symmetric quartics
by just introducing the x squared term into the equation as well.
So if we, for example, look here at the graph
of y = x to the power of 4 – 3x squared,
we see a nice symmetric shape with three turning points
and three x-intercepts.
Obviously, altering 'e' would simply translate the graph up and down.
So we could take this same shape and translate it up or down,
hence also being able to create a quartic shape
with three turning points and four x-intercepts.
Similarly, three turning points and no x-intercepts
is quite possible as well.
Making 'a' negative here would obviously reflect this shape.
We would need to also make 'c' positive there and make 'a' negative
to get the three turning points occurring.
Alright, so putting this back to one.
And then we can have many different quartic functions
which aren't symmetric.
So we might, for example ... Let's move that back to zero as well.
We might, for example, have something like this,
where, in fact, we have a quartic function
still with only one turning point but with two x-intercepts,
and this sort of slightly strange asymmetric shape.
And that can obviously go the other way as well.
We can shift it up and down
and we could reflect it and still maintain that same kind of shape.
Sticking with the asymmetric vein,
we could also create an asymmetric shape
simply by adding in an x cubed term,
and we see here we get only two stationary points,
one turning point and one point of inflection,
and also only two x-intercepts.
But again, translation up or down.
We can affect what's going on
and maintain that same shape, but in a different position.
And the last kind of shape we might get
would be returning to the 'W' sort of shape,
but with some asymmetry involved.
So if I also add in a 'c' term here, for example ...
... you'll see -
let me try making 'a' bigger as well so we can still see that there -
you'll see we're starting to get an extra actual turning point
come out of that point of inflection in our previous example,
so we now have three turning points occurring there
and hence three x-intercepts.
Similarly, a shift up or down could make that four x-intercepts.
And so we get lots of different shapes that we could have.
We can get symmetric sort of bowl shapes
with just one stationary point,
we could get symmetric 'W' shapes with three stationary points,
we could get asymmetric shapes
with just the one turning point, one stationary point,
we could get asymmetric 'W's,
we could get asymmetric shapes with one turning point
and one point of inflection ...
So there are many different combinations that could occur
and 'a', 'b', 'c', 'd' and 'e' here are all quite entwined,
although we can continue to link 'a' with the dilations and reflections
and 'e' with the translations up and down.
So there's a snapshot of what can occur with quartic functions.