I'm always on the look out for interesting mathematical 'nuggets' that might lead to something suitable for a Maths HL or SL student Exploration (IA task). Recently, I came across the following fact about parabolas that I was not aware of and thought was very intriguing.

So, if the distance h between the two points where the secant line intersects the parabola is kept constant for a particular parabola $y=a\cdot x^2$ then the area of the region bounded by the parabola and the secant line should remain constant.

I consider this an interesting mathematical 'nugget' that has the potential to expand into a topic that a student could explore. I thought it would be fun to do some exploring myself - and this fact about parabolas is very appropriate for confirming with a dynamic graph. Recently I've been playing around with Desmos, the free online graphing calculator. Below is a short video (30 sec) showing my Desmos graph in motion - with the secant line moving but the horizontal distance between the intersection points is always 4 units. Since Desmos does not have a built-in function for computing a definite integral I entered an expression to add 500 rectangles between the graphs from the left intersection point to the right intersection point to compute the area of the shaded region.

The expressions list for this Desmos graph (the user-defined items on the left side that control various aspects of the graph) is shown at right.  The parabola used in this demonstration is $f\left(x\right)=\frac{1}{4}{x}^2$ and $g\left(x\right)$ is the equation for the secant line where p is the x-coordinate of the left intersection point and h is the horizontal distance to the right intersection point. For this graph the value of p is -1 and h is 4.

One of the nicest features of constructing a graph with Desmos is that it is so easy to make it dynamic. As soon as a constant is entered the program asks if you wish to make a slider so that you can easily change the value of the constant. So all of the constants contained in this graph - a, p, h and n - can be changed manually by dragging the slider or changed automatically by pressing the 'animate' button - and then the graph will be animated according to the changing parameter values. The video above is a recording of the animation created after pushing the animate button for p.

Remember, according to the statement with which I started this blog post, the area of the bounded region is supposed to be dependent only on the value of a and h.  By playing around with the values for a and h and making note of the area A, one may be able to conjecture a formula for A in terms of a and h (not easy).

Coming up with the correct formula and proving it would be a good start for an Exploration but would probably not be enough for an Exploration that would earn high marks. In order to have a good chance at writing a strong Exploration, I would encourage a student to devise their own further extensions to this initial investigation. For example, one could consider other parabolas that are not in the form $y=a\cdot x^2$ - for example, $y=a{x}^2+bx+c$ or $y=a{\left(x-h\right)}^2+k$ .  And perhaps consider if a similar property exists for other polynomial functions (e.g. cubic, quartic), or consider whether a similar property occurs in three dimensions.