Text description - screencast for interactive 1

[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're looking at graphs of the form y equals a times (x minus h) cubed plus k. So here we have three constants, a, h and k, which are going to affect the graph of y equals 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 equals 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 equals 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 equals 0.
And now, looking at negative values 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 equals -3x cubed and this is the graph of y equals 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 minus 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 minus 3) all cubed and this graph has moved to the right by three. (x minus 4) all cubed and we've moved to the right by four.
If we make h negative... Here we're looking at (x minus negative 3) all cubed, so that is (x plus 3) all cubed, which has gone to the left by three. Here we're looking at (x minus negative 5) all cubed, so hence (x plus 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 equals x cubed plus 2 and this is the graph of y equals x cubed translated up by two. As we reduce the size of k... We're now looking at the graph of y equals x cubed minus 3 and we see that this is the graph of y equals 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 plus 2) all cubed has gone to the left by two, which seems a little counterintuitive, and (x minus 2) all cubed goes to the right by two. K affects horizontal - sorry - vertical translations and they are less counterintuitive in that x cubed plus 2 in fact goes up by two and x cubed minus 2 does indeed go down by two. So a is dilations and reflections of the graph, h and k translations.