Text description - screencast for interactive 4

[In this video, an electronic x and y graph is used to show how equations can be presented in graph form. Below the graph, adjustable sliders allow individual values in the equation to be changed. As the values in the equation are changed, the shape of the graph changes. The narrator describes the changes in the graph.]

NARRATOR: In this interactive, we will explore graphs of quartic functions, that is, quartic functions with equations of the form y equals a times x to the power of four plus bx cubed plus cx squared plus dx plus 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 equals 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 equals x to the power of 4 minus 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 minus let me try making a bigger as well so we can still see that there minus 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.