Text description - screencast for interactive 3
[In this video, two electronic x and y graphs are used to show how equations can be presented in graph form. In the graph on the left, an adjustable pointer allows individual values in the equations to be changed. As the values in the equations are changed, the shape of the right-hand graph changes. One section of the left-hand graph is shaded. The narrator describes the changes in the graphs.]
NARRATOR: In this interactive we consider cubic graphs of the form y equals x cubed plus bx squared plus cx.
In this interactive we're able to change the values of b and c using this graph in the left-hand side by selecting pairs of values of b and c relative to the equation b squared minus 3c equals 0.
So first I'd like to think about where that equation b squared minus 3c equals 0 comes from. So if we look at graphs of the form y equals x cubed plus bx squared plus cx and we consider their stationary points, we know we do that by solving the derivative equal to zero. So the derivative is 3x squared plus 2bx plus c and that equal to zero.
We now have a quadratic equation, and if we're looking at the number of stationary points that we might get from cubic graphs of this form, we'd be looking at the discriminant of this quadratic equation, and the discriminant works out to be 4b squared minus 12c. So we know that equations of this form would have two stationary points if 4b squared minus 12c is bigger than zero, and that is if b squared minus 3c is bigger than zero, we'd have one stationary point if b squared minus 3c is equal to zero, and no stationary points if b squared minus 3c is less than zero.
So in fact, what we're seeing in this interactive is a way to manipulate the number of stationary points that are cubic. So we should find if we select points on the line b squared minus 3c, we should be getting cubic graphs with just one stationary point. If we select points that are within this shaded region, that is, where b squared minus 3c is less than zero, we see that we're getting no stationary points, which agrees with the algebra that we saw earlier. And lastly, if we select points in the region where b squared minus 3c is bigger than zero, we see that we do indeed get cubic graphs with two stationary points.